Logarithms
Natural logarithms
None
Natural logarithm with a calculator
Natural Logarithm with a Calculator
Discussion and questions for this video
 Use a calculator to find log base e of 67
 to the nearest thousandth.
 So just as a reminder, e is one of these crazy numbers
 that shows up in nature, in finance, and all these things,
 and it's approximately equal to 2.71
 and it just keeps going on and on and on.
 So you could view log base e as 67.
 Let's see, what does e mean? e is just a number,
 just like pi is just a number.
 So this is really the same thing as saying log base 2.71,
 and the actual numbers, so you'd have
 to write all the digits that keep on going forever
 and never repeat 6 of 67.
 So what power do I have to raise e to to get to 67?
 So another way of saying that is this is equal to x.
 You're saying e to the x is equal to 67,
 we need to figure out what x is.
 Now, traditionally you will never
 see someone write log base e even though e
 is one of the most common bases to take a logarithm of.
 And so the reason why you wouldn't see log base e written
 this way is log base e is referred
 to as the natural logarithm.
 And I think that's used because e shows up so many times
 in nature.
 So log base e of 67, another way of saying that or seeing
 that, and the more typical way of seeing
 that is the natural log.
 And I think this is ln, so I think
 it's maybe from French or something, log natural, of 67.
 So this is the same thing as log base e of 67.
 This is saying the exact same thing.
 To what power do I have to raise e to to get 67?
 When you see this ln, it literally means log base e.
 Now, they let us use a calculator,
 and that's good because I don't know off the top of my head
 what power I have to raise 2.71 so on and so forth what
 power I have to raise that to to get to 67.
 So we'll get our calculator out.
 So we get the TI85 out.
 And different calculators will have
 different ways of doing it.
 If you have a graphing calculator like this,
 you literally can literally type in the statement natural log
 of 67 then evaluate it.
 So here this is the button for ln,
 means natural log, log natural, maybe.
 ln of 67, and then you press Enter,
 and it'll give you the answer.
 If you don't have a graphing calculator,
 you might have to press 67 and then press natural log
 to give you the answer, but a graphing calculator
 can literally type it in the way that you would write it out,
 and then you would press Enter.
 So 4.20469 and we want to round to the nearest thousandth.
 So this is the thousandths place right here, this 4.
 The digit after that is 5 or larger, it's a 6,
 so we're going to round up.
 So this is 4.205.
 So this is approximately equal to 4.205.
 And it actually makes a lot of sense,
 because we know that e is greater than 2,
 and it is less than 3.
 And if you think about what 2 to the fourth power
 gets you to 16.
 And 3 to the fourth power gets you to 81.
 67 is between 16 and 81 and e is between 2 and 3.
 So at least it feels right that's
 something that's like 2.71 to the little
 over the fourth power should get you to a number that's
 pretty close to 3 to the fourth power.
 Actually that makes sense because it's actually
 closer to 3.
 2.71 is closer to 3 than it is to 2.
 So this feels right, that you take this to the fourth,
 little over the fourth power, you get to 67.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

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So does anyone know if he was right about that "log natural" french thing?
It's actually written "ln" instead of "nl" because the Latin name of natural log is "logarithmus naturali."
I was wondering the same too! I think he's wrong! But in french, we actually call this log "le logarithme népérien" and it's name after a Scottish mathematician John Neper working on logarithm tables, here's the wikipedia in english: https://en.wikipedia.org/wiki/John_Napier
Logarithmus naturali is how I was taught it too, latin for natural log.
I was taught log natural by my high school teacher...
..well I guess he's right in the fact that we place the term "logarithme" before "népérien", further abbreviated "ln" !
logarithmus naturalis is the full name but yes
It's called logarithmus naturalis in latin.
We actually called it the "logarithme naturel" ;p
If I am looking for a calculator to help me with this should I get a scientific calculator or graphing calculator?
I would say graphing. Either a Texas Instruments or a Casio, depending on what your school division uses (if applicable). If that's not a factor, Casios are more powerful, and some of the operations are more intuitive, but most instructions that I have found tend to be written for the TI series  and it's what Khan uses. Graphing calculators do everything that a scientific calculator does, and much more. Because of that, however, they tend to be a fair amount more expensive. If you plan on continuing in any math education, get the graphing calculator. Scientific calculators are just too limiting for higherlevel math.
The TI84 Plus Graphing Calculator is an excellent choice. I have one that allow you to change the base of an log without actually doing oddball math. It may need an update the the operating system, though.
The TI series of graphing calculators are an excellent brand of calculators. I personally use the TI83plus, because it is cheaper than the TI84 and not a whole lot different than the newer version (besides being able to change the base of a log. If that's important to you, than consider the TI84plus, but its more $$$). As for scientific calculators, you should definitely have at least one because they are so much easier to operate than graphing calcs. Texas Instruments sells a variety of these simpler (but very useful) devices. For a final note, I wouldn't buy more than one graphing calculator  often computers can do the same things graphing calculators can, but even faster. Because of this, graphing calculators are more for students, whereas adults who need to do that sort of stuff usually use their laptops for the job. This reply might have come a bit late for jtfeliz, but I hope it shines some light on selecting a calculator that's right for you for anyone else who needs a calculator.
Where can I get the calculator Sal is using but for mac?
I would have to agree with CasualJames. The biggest benefit that a graphing calculator has is the ability to create tables, and graphs which makes it easier to recheck your answers. I would recommend the TI84 plus, because it has many more operations that are much more userfriendly.
Both will be useful...but a scientific calculator is better in the long run....and its easier to use.
Where and how does "e" appear in nature?
e=(1+(1/x))^x The variable "e" is often used in calculating equations in physics, such as Newton's Law of Cooling (look it up, I won't explain it as well as others will). It is also used in finance with compound interest. Mr. Khan probably has something on it in his finance videos.
"e" appears in nature in the design of flowers, and the placement of designs on turtle shells and other such natural geometric shapes.
see the compound interest video, he explains e there
When Sal says e shows up in nature a lot, what does he mean? Where in nature does e show? Like, I know pi, in nature, is the ratio of circumference to diameter, is there any such thing for e?
In my work, I encountered e a lot more than π. The constant shows up in exponential functions all the time, such as in radioactive decay. It even shows up in such things as statistics, business math, civil engineering, and computing interest  just to name a few.
As far as why we would use such a strange number as e as the preferred base for a logarithm, that will become very evident if you go on to study calculus. For now, let us just say that the math is so much easier with the natural logarithm that in higher levels of math and many applied uses of math, the natural log is used almost to the exclusion of any other base. There are some specialized fields where it makes more sense to use a base 2 or base 10 logarithm, but the natural log is far, far easier in the vast majority of applications.
As far as why we would use such a strange number as e as the preferred base for a logarithm, that will become very evident if you go on to study calculus. For now, let us just say that the math is so much easier with the natural logarithm that in higher levels of math and many applied uses of math, the natural log is used almost to the exclusion of any other base. There are some specialized fields where it makes more sense to use a base 2 or base 10 logarithm, but the natural log is far, far easier in the vast majority of applications.
What if you have a number in front of the e instead of log? Like...4e^x = 10? How would you go about doing that?
Taylor,
Try dividing both sides by 4 first.
e^x = 10/4
Now can you solve?
Try dividing both sides by 4 first.
e^x = 10/4
Now can you solve?
For problems that add/subtract to/from the x, simply solve for the exponent by using ln. In the example you gave:
e^(x4) = 2
x  4 = ln(2)
x = ln(2) + 4
An example for division:
e^(x/5) = 2
Same thing as before.
Use the ln.
x/5 = ln(2)
x = 5 ln(2)
For your last example let's equate it to some constant just for the sake of clarity. We'll choose 2 because it's a really friendly number:
e^(ln(5x)) = 2
Now, we have an important identity for logs. x^(log(N) base x) = N
so e^(ln(5x)) = 5x
so 5x = 2 and finally x = 2/5 Hope this helps. :)
e^(x4) = 2
x  4 = ln(2)
x = ln(2) + 4
An example for division:
e^(x/5) = 2
Same thing as before.
Use the ln.
x/5 = ln(2)
x = 5 ln(2)
For your last example let's equate it to some constant just for the sake of clarity. We'll choose 2 because it's a really friendly number:
e^(ln(5x)) = 2
Now, we have an important identity for logs. x^(log(N) base x) = N
so e^(ln(5x)) = 5x
so 5x = 2 and finally x = 2/5 Hope this helps. :)
Yeah. You would divide 10 and 4, and get 2.5, and then the final answer would be x = 0.92 (rounded to the hundredth place), right?
What about if you had e^x4 = 2? Or any other problem that adds to the x, divides the x by a number, or does e^ln 5x? I have a test on Friday on all of this stuff, and I'm completely clueless.
What about if you had e^x4 = 2? Or any other problem that adds to the x, divides the x by a number, or does e^ln 5x? I have a test on Friday on all of this stuff, and I'm completely clueless.
Where does "e" come up in nature? Just curious.
"e" is the natural representation for any problem involving exponential growth. For example, halflife problems are typically expressed at the college level using "e", as it gives you a clean connection between the amount of the radioactive substance remaining and the current rate of decay (the level of radiation).
hi my name is justin
What is the difference b/w log and ln??
ln is the natural logarithm. It is log to the base of e.
e is an irrational and transcendental number the first few digit of which are:
2.718281828459...
In higher mathematics the natural logarithm is the log that is usually used. The log on your calculator is the common log, which is log base 10.
e is an irrational and transcendental number the first few digit of which are:
2.718281828459...
In higher mathematics the natural logarithm is the log that is usually used. The log on your calculator is the common log, which is log base 10.
If e is truly infinite, how can one tell that it _never_ repeats? You'll never reach the end of the number, and thus, you won't now the number's true value to know if it repeats.
A number that repeats digits in its decimal form is a rational number meaning that it can be represented as the quotient of two whole numbers. There is a high level proof on this link that e is irrational: http://en.wikipedia.org/wiki/Proof_that_e_is_irrational
e is not infinite, it is a finite number. However, when expressed in decimal form, it has infinitely many digits.
As Andrewp18 correctly points out, some of the proofs for numbers being irrational or transcendental require quite high level mathematics.
The usual way to prove a number is irrational is by what is known as "proof by contradiction". To do this, you assume the opposite, that the number is rational, and then you prove this assumption leads to a contradiction, thus it cannot be true. Thus, you have proved the number is irrational by proving it cannot be rational. There is no need to know all of the digits or even very many of the digits to know whether a number is irrational.
As Andrewp18 correctly points out, some of the proofs for numbers being irrational or transcendental require quite high level mathematics.
The usual way to prove a number is irrational is by what is known as "proof by contradiction". To do this, you assume the opposite, that the number is rational, and then you prove this assumption leads to a contradiction, thus it cannot be true. Thus, you have proved the number is irrational by proving it cannot be rational. There is no need to know all of the digits or even very many of the digits to know whether a number is irrational.
Does anyone know how to change the base on a TI83 to something other than ten?
actually ON ALL CALCULATORS you have to do this equation, log a(b) = log b(b)/log b(a)
Sal made another video on how to calculate different base logarithms with a basic calculator. Log base(x) of (y) equals log(y)/log(x).
Thank you CasualJames I was trying to find the formula, but with limited resources that is a little difficult.
To echo the previous answers, you can't on an 83.
The change of base formula, incidentally, is log a(b) = log b(b)/log b(a)
The change of base formula, incidentally, is log a(b) = log b(b)/log b(a)
Ryan Farias is right TI83 cannot change the base without doing a different formula. I am confused as to what that formula is though... I know that it has something to do with dividing the base and/or the log by one or the other.
You cant sorry your calculator doesnt have it programed but give me problem your having trouble with i can help. =)
I'm not sure about TI83, but on TI84 plus, I know this works:
Press MATH
Scroll down the MATH list until to see logBASE(
Press ENTER
This will give you the LOG(base A of B) function.
Let me know if it works on TI83! Hope it helps! =)
Press MATH
Scroll down the MATH list until to see logBASE(
Press ENTER
This will give you the LOG(base A of B) function.
Let me know if it works on TI83! Hope it helps! =)
Where exactly is the number "e" found in nature? Why do people call it a natural number?
e comes up all the time in realworld math. For example, it is used in business math for certain kinds of interest calculations. It is used in calculating certain kinds of reaction rates, especially radioactive decay. It is used extensively in engineering computations.
What if you have a question that asks like e^2x+1=55? How can you solve for e?
Ruaida,
"e" is a transcendental number, kind of like "pi." So you don't need to solve for it. We already know what 'e' is. It's about 2.71828182845….
What you probably want to solve for is "x". Here is how you would do that:
(e^2x) +1=55. (I added the parenthesis around e^2x just to make it a little more clear) Subtract 1 from both sides:
e^2x=54. Take the natural logarithm of both sides (that's 'ln' on your calculator usually, not 'log')
2x=ln 54. Divide both sides by 2.
x=(1/2)ln (54) (By the way, this is the same as x=ln (54^1/2) or ln (sqrt(54)).
x= 1/2 *3.98898… = 1.9945 (approximately).
"e" is a transcendental number, kind of like "pi." So you don't need to solve for it. We already know what 'e' is. It's about 2.71828182845….
What you probably want to solve for is "x". Here is how you would do that:
(e^2x) +1=55. (I added the parenthesis around e^2x just to make it a little more clear) Subtract 1 from both sides:
e^2x=54. Take the natural logarithm of both sides (that's 'ln' on your calculator usually, not 'log')
2x=ln 54. Divide both sides by 2.
x=(1/2)ln (54) (By the way, this is the same as x=ln (54^1/2) or ln (sqrt(54)).
x= 1/2 *3.98898… = 1.9945 (approximately).
do you mean solve for x? since e is already quantified like π, it'd be a bit strange to solve for e. If you meant x; follow this approach, take a ln of each side so that you get lne^2x + ln1= ln55
then use log rules, to get 2xlne+ln1=ln55, then solve for x. where x= (ln55ln1)/2... also just in case you wondered where lne went, its actually equal to 1. hope that helps a bit
then use log rules, to get 2xlne+ln1=ln55, then solve for x. where x= (ln55ln1)/2... also just in case you wondered where lne went, its actually equal to 1. hope that helps a bit
What is e and why is it used so much, and where? I know it is like pi, both are just numbers that go on forever. Pi is about 3.1415, and it is used with circles. You need pi to find the area and circumference.
E is about 2.71, but where is e used and why is it so commonly used as a base for logarithms?
Thanks.
E is about 2.71, but where is e used and why is it so commonly used as a base for logarithms?
Thanks.
`e` or `Euler's number` is a very important mathematical constant that is found in many things. It estimates to be about `2.71828`... It is the base of natural logarithms. `e` is used in calculus, complex numbers, chemistry and much more.
You can use a binomial expansion to calculate `e`. For example take a look at `(1 + 1/n)^n` if this `n` approaches infinity or is extremely large it approaches the value of `e`. If n = 100000 then the output is `e` correctly only to the 4th decimal place.
`e` is the limiting factor used in all kinds of scientific processes and economic functions.
You can use a binomial expansion to calculate `e`. For example take a look at `(1 + 1/n)^n` if this `n` approaches infinity or is extremely large it approaches the value of `e`. If n = 100000 then the output is `e` correctly only to the 4th decimal place.
`e` is the limiting factor used in all kinds of scientific processes and economic functions.
Can e be represented by an equation?
Yes it can! e^x = 1 + x + x^2/2 + x^3/3! + x^4/4!..... or you can express it as lim n>infinity (1+1/n)^n
where does e appear in nature?
`e` or `Euler's number` is a very important mathematical constant that is found in many things. It estimates to be about `2.71828`... It is the base of natural logarithms. `e` is used in calculus, complex numbers, chemistry and much more.
You can use a binomial expansion to calculate `e`. For example take a look at `(1 + 1/n)^n` if this `n` approaches infinity or is extremely large it approaches the value of `e`. If n = 100000 then the output is `e` correctly only to the 4th decimal place.
`e` is the limiting factor used in all kinds of scientific processes and economic functions.
You can use a binomial expansion to calculate `e`. For example take a look at `(1 + 1/n)^n` if this `n` approaches infinity or is extremely large it approaches the value of `e`. If n = 100000 then the output is `e` correctly only to the 4th decimal place.
`e` is the limiting factor used in all kinds of scientific processes and economic functions.
So I have this assignment where I need to condense an expression with two natural logarithms...
2 ln 7  3 ln 4.... I don't get it. At all.
2 ln 7  3 ln 4.... I don't get it. At all.
It should work the same way it does with log. You could put the multipliers (2 for the first term, 3 in the second) within the log operation as an exponent of the quantity inside the logarithm.
If you start with a*log(x) that equals log(x^a) as demonstrated in the logarithm set of videos.
Another property is that log(x)log(y)=log(x/y), also demonstrated in the previous section.
Since ln is just log at base e instead of base 10, it responds to the properties identically.
If you start with a*log(x) that equals log(x^a) as demonstrated in the logarithm set of videos.
Another property is that log(x)log(y)=log(x/y), also demonstrated in the previous section.
Since ln is just log at base e instead of base 10, it responds to the properties identically.
how do you go on solving problems like this 4^−9t = 0.60 ... it's not an homework question I am just puzzled what ln is or if it's a problem like this where do you plug it in?
ln, the natural logarithm, is log base e ( an irrational number, not a variable). It is important in calculus. I do not think that it would help you with 4^9t = 0.6, though. Here is how:
4^9t = 0.6
9t*log(4) = log(0.6)
t*(9*log(4)) = log(0.6)
t= log(0.6)/(9*log(4))
4^9t = 0.6
9t*log(4) = log(0.6)
t*(9*log(4)) = log(0.6)
t= log(0.6)/(9*log(4))
When you have the variable you're trying to solve in an exponent, you want to "bring it down to normal level". As you point out, ln can help you a lot:
4^−9t = 0.60
ln (4^(−9t)) = ln 0.60 [take the natural log of both sides]
9t ln 4 = ln 0.60 [use log property logₓyⁿ = nlogₓy {x is the base which in our case is e}]
ln 4 and ln 0.60 are just constants so solving for t should now be straightforward:
9t = (0.5108...)/(1.3862...) = 0.3684...
=> t = 0.0409...
Wolfram Alpha confirms our work: http://www.wolframalpha.com/input/?i=4%5E%28%E2%88%929t%29+%3D+0.60
4^−9t = 0.60
ln (4^(−9t)) = ln 0.60 [take the natural log of both sides]
9t ln 4 = ln 0.60 [use log property logₓyⁿ = nlogₓy {x is the base which in our case is e}]
ln 4 and ln 0.60 are just constants so solving for t should now be straightforward:
9t = (0.5108...)/(1.3862...) = 0.3684...
=> t = 0.0409...
Wolfram Alpha confirms our work: http://www.wolframalpha.com/input/?i=4%5E%28%E2%88%929t%29+%3D+0.60
What is an mathematical explation of what e stands for?
ACTUALLY ↑←hkapur97→↓ e^iπ =1 and e=lim_x→∞(1+1/x)^x
No, e has several definitions. Here is 1 
e = limit as n approaches infinity of (1 +1/n)^n
Go to the precalculus playlist, and watch the videos on compound interest and e
e = limit as n approaches infinity of (1 +1/n)^n
Go to the precalculus playlist, and watch the videos on compound interest and e
If you have a dollar and you have an interest rate of 100% that is compounded an infinite amount of times every second then at the end of the year you will have e dollars which is about 2.71828183
e^x is f(x) when (f´(x))/(f(x)=1
not sure if that was that helpful
not sure if that was that helpful
so that means the iπ√1=e and log_e〖1〗=iπ
look it up on Wikipedia, you'll get a really good definition there.
What I'm more interested in is how to calculate this WITHOUT a calculator
To compute a logarithm (except for simple cases where it is an integer or easy fraction) without a calculator or a table of logs, is very difficult. I can show you the calculation, though:
ln(x) = ∑(n=1 to ∞) 2*[(x1)/(x+1))]^(2n1)/(2n1)
Doing this for ln(4) we get:
ln (4) = 6/5 + 18/125 + 486/15625 + 4374/546875 + 4374/1953125 ....
The first 5 elements of the series sum to about 1.38534...
The actual ln (4) = 1.38629...
To get very many digits correct, you would need to add the first 20 or 30 members of the series.
ln(x) = ∑(n=1 to ∞) 2*[(x1)/(x+1))]^(2n1)/(2n1)
Doing this for ln(4) we get:
ln (4) = 6/5 + 18/125 + 486/15625 + 4374/546875 + 4374/1953125 ....
The first 5 elements of the series sum to about 1.38534...
The actual ln (4) = 1.38629...
To get very many digits correct, you would need to add the first 20 or 30 members of the series.
Unfortunately, it's not very practical. Unless you have somethin really simple, like ln(e^3) = 3, It's impossible to work out a logarithm without a calculator, unless you want to spend hours doing trial and error.
I was wondering, when considering the expression ln(2e)^2, does the the square belong to the "ln" function as a whole or does it belong to the "2e" separately?
To the whole: `(ln(2e))^2`
I didn't see that formula in the video. I would have to see the original context to be sure something wasn't lost in the translation to text.
I didn't see that formula in the video. I would have to see the original context to be sure something wasn't lost in the translation to text.
To the logarithm's argument. You would write that squared logarithm as `ln^2 (2e)`.
You can simplify this using the property of logs:
(A) `ln(xy) = ln(x) + ln(y)`
(B) `ln(x^a) = a*ln(x)`
This is covered in the "Logarithm basics" section, which you may want to review. (Note that B can be derived from A, but it happens often enough to memorize.)
Looking at the whole formula, I think the parenthesis are misplaced. It should be:
`ln(2e^2)`
which expands to:
`ln(2)+ln(e)+ln(e)`
or
`ln(2)+2*ln(e)`
The other terms can be expanded in the same way. Note that 4 and 8 are powers of 2. The 1/2 exponent can be simplified using (B), or by using the properties of exponents. If you work it both ways, you will see why logs are cool.
Please keep asking if you need more help.
(A) `ln(xy) = ln(x) + ln(y)`
(B) `ln(x^a) = a*ln(x)`
This is covered in the "Logarithm basics" section, which you may want to review. (Note that B can be derived from A, but it happens often enough to memorize.)
Looking at the whole formula, I think the parenthesis are misplaced. It should be:
`ln(2e^2)`
which expands to:
`ln(2)+ln(e)+ln(e)`
or
`ln(2)+2*ln(e)`
The other terms can be expanded in the same way. Note that 4 and 8 are powers of 2. The 1/2 exponent can be simplified using (B), or by using the properties of exponents. If you work it both ways, you will see why logs are cool.
Please keep asking if you need more help.
Thanks so much guys :), this makes a huge difference. The original question was, " Simplify ln(2e)^2 + ln((4e^6)^0.5)  ln(8e^7) without the use of a calculator."
Ln actually mean "Logarithm Népérien" reffering to John Napier, who discovered Logarithms. http://en.wikipedia.org/wiki/John_Napier
Can someone give an explanation of e? What is its significance? Who discovered it? (etc.) Or, if there is a video I did not happen to notice on this subject, can someone direct me to a tutorial? Thanks!
Euler's number, commonly known as 'e', is no less important than Pi or any other mathematical constant. The only thing is that we don't bump into it as often, at least while we're still in high school, or pursuing a career in a completely different field.
The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact e): (http://upload.wikimedia.org/math/5/d/1/5d18070ef4fdc2e1ae46d38ad588b18a.png)
Its significance? There are many. Some of them are:
1) "e" is the base of Natural Logarithm (ln). Given that x=ln (y), then y = e^x
2) "e" is also used in complex numbers, e^(ix) = cos x + i sin x.
3) "e" has a special place in Calculus, d/dx (e^x) = e^x, d/dx (ln x) = 1/x
4) "e" is also used in the definitions of hyperbolic functions, sinh x , cosh x and tanh x
5) "e" is the limit of (1 + 1/n)^n as n approaches infinity, an expression that arises in the study of compound interest.
You may want to check out this Wikipedia article for more info:
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
Hope this helps! :)
The discovery of the constant itself is credited to Jacob Bernoulli, who attempted to find the value of the following expression (which is in fact e): (http://upload.wikimedia.org/math/5/d/1/5d18070ef4fdc2e1ae46d38ad588b18a.png)
Its significance? There are many. Some of them are:
1) "e" is the base of Natural Logarithm (ln). Given that x=ln (y), then y = e^x
2) "e" is also used in complex numbers, e^(ix) = cos x + i sin x.
3) "e" has a special place in Calculus, d/dx (e^x) = e^x, d/dx (ln x) = 1/x
4) "e" is also used in the definitions of hyperbolic functions, sinh x , cosh x and tanh x
5) "e" is the limit of (1 + 1/n)^n as n approaches infinity, an expression that arises in the study of compound interest.
You may want to check out this Wikipedia article for more info:
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
Hope this helps! :)
can someone help me with this equation please
2ln x + ln 5 = 7.2
2ln x + ln 5 = 7.2
How do you make the radical or square root sign on your computer?
it will become
ln x^2(5)= 7.2
e^7.2=x^2(5)
1339.4308=x^2(5)
1339.4308/5=x^2(5)/5
√267.8862=√x^2
16.3672=x
ln x^2(5)= 7.2
e^7.2=x^2(5)
1339.4308=x^2(5)
1339.4308/5=x^2(5)/5
√267.8862=√x^2
16.3672=x
can anyone help me with this equation
e^x5e^x=4
e^x5e^x=4
e^x5*e^(x)=4>e^x5*(1/(e^x))=4>e^x5/(e^x)=4. Let e^x be n.
n5/n=4 (Multiply each side by n)>n^2+5=4n (Subtract each side by 4n)>n^24n+5=0 (Factor)>(n5)(n+1)=0.
n5=0 or n+1=0, thus, n=5 or n=1. Now, we substitute e^x for n: e^x=5 or e^x=1. Take the natural log of each side: ln(e^x)=ln(5) or ln(e^x)=ln(1) (Simplify)>x=ln(5) or x=ln(1).
The natural log of a number is the number of times you would have to multiply e to get that number. However, no matter how many times you multiply e, you can't get a negative number. So, our answer is ln(5), which is approximately 1.60943791.
n5/n=4 (Multiply each side by n)>n^2+5=4n (Subtract each side by 4n)>n^24n+5=0 (Factor)>(n5)(n+1)=0.
n5=0 or n+1=0, thus, n=5 or n=1. Now, we substitute e^x for n: e^x=5 or e^x=1. Take the natural log of each side: ln(e^x)=ln(5) or ln(e^x)=ln(1) (Simplify)>x=ln(5) or x=ln(1).
The natural log of a number is the number of times you would have to multiply e to get that number. However, no matter how many times you multiply e, you can't get a negative number. So, our answer is ln(5), which is approximately 1.60943791.
How would you solve y =ln( x3/5  8) for x?
Start with the outermost functions, and 'undo' them on each side. The 'opposite' of the ln function is the exp function, so we have exp[y] = exp[ln(x^3 / 5  8)], and therefore exp[y] = x^3 / 5  8. Now add 8 to both sides: exp[y] + 8 = x^3 / 5. Multiply both sides by 5: 5exp[y] + 40 = x^3. Now cuberoot both sides: (5exp[y] + 40)^(1/3) = x, and you've solved for x.
Thankx! but somehow ur answer isnt the same of my teachers! This is how he solved it:
y =ln(x3/58)
e^y=x^3/58
e^y+8=x^3/5
(e^y+8)^5/3=x
x=(e^y+8)^5/3
Could u explain this?
y =ln(x3/58)
e^y=x^3/58
e^y+8=x^3/5
(e^y+8)^5/3=x
x=(e^y+8)^5/3
Could u explain this?
Here in the question in the above video , what does that nearest thousand means ?
Just to add to the already good answer above by Ammara Khan,, a thousandth is 3 decimals. How come? Well, one tenth is 1 decimal, because one tenth is 0,1. Maybe this is what you mean with your question,
How would I go about finding the number whose natural log is a given number? e.g. 2.76
So really my question is, how would I work out 'x' of: ln x = 2.76
This is not covered in this section that I can see?
So really my question is, how would I work out 'x' of: ln x = 2.76
This is not covered in this section that I can see?
```ln(x) = 2.76
e^ln(x) = e^(2.76)
x = e^(2.76)```
e^ln(x) = e^(2.76)
x = e^(2.76)```
How do you find natural logs without a calculator?
Unless you are very advanced in your math skills, the short answer is that you don't.
The standard custom at this level of study is either express your answer in terms of the natural log, to use a calculator or other computing device, or you will be given the logs you need.
There is a way to solve natural logs by hand, of course, but it is quite difficult math and you will not be asked to do it at this level of study. The only exception would be that you might be expected to recognize those special cases where the natural log has a nice, easy solution (such the natural log of e raised to some exponent).
But, just for reference, the actual computation is:
ln x = lim h→0 [x^(h) − 1] / h
There are other ways to compute the natural logarithm by hand, none of them easy.
The standard custom at this level of study is either express your answer in terms of the natural log, to use a calculator or other computing device, or you will be given the logs you need.
There is a way to solve natural logs by hand, of course, but it is quite difficult math and you will not be asked to do it at this level of study. The only exception would be that you might be expected to recognize those special cases where the natural log has a nice, easy solution (such the natural log of e raised to some exponent).
But, just for reference, the actual computation is:
ln x = lim h→0 [x^(h) − 1] / h
There are other ways to compute the natural logarithm by hand, none of them easy.
At 2:55, I cannot understand how does ln(67)=4.205 approximately matches between 2 and 3. plz help:(
If we let ln67 = a
then e^a = 67 by definition of a logarithm
e by definition is about 2.71 I think
If we round the 4.2 down for approx.
2^4 = 16 and 3^4 = 81
Therefore since e = 2.71 which is in between 2 and 3
we expect e^4 to be in between 16 and 81, which it is.
then e^a = 67 by definition of a logarithm
e by definition is about 2.71 I think
If we round the 4.2 down for approx.
2^4 = 16 and 3^4 = 81
Therefore since e = 2.71 which is in between 2 and 3
we expect e^4 to be in between 16 and 81, which it is.
Can someone prove this for me?
lim (a^x 1)/x = ln(a)
x>0
Thanks. :)
lim (a^x 1)/x = ln(a)
x>0
Thanks. :)
Let me know if you want this proved, but the derivative of aᵡ = aᵡ · ln(a) for a > 0.
First, let's see what happens if we just plug x=0 into our expression:
(a⁰1) / 0 = (11) / 0 = 0 / 0. This is undefined and we can use L'Hôpital's rule (check out the link if this is unfamiliar to you). L'Hôpital's rule states that if a limit evaluates to 0/0 (or (±∞)/(±∞), I believe), you can evaluate the limit of the derivative of the nominator divided by the derivative of the denominator (differentiate them both independently). Let's try that:
1. Differentiate both the numerator and the denominator of our expression:
• d/dx [aᵡ  1] = aᵡ · ln(a)  0 = aᵡ · ln(a)
• d/dx [x] = 1
2. Rewrite the original expression as (aᵡ  1)' / x':
(aᵡ · ln(a)) / 1 = aᵡ · ln(a)
3. Reevaluate the limit (let's plug in x=0 once again. If you still end up with 0/0 or (±∞)/(±∞), you can keep on using the rule until you eventually end up with something else):
Plug in x=0:
a⁰ · ln(a) = 1 · ln(a) = ln(a)
We have now proved that the limit of (aᵡ  1) / x = ln(a) as x approaches 0.
An introduction to L'Hôpital's rule:
https://www.khanacademy.org/math/calculus/derivative_applications/lhopital_rule/v/introductiontolhopitalsrule
First, let's see what happens if we just plug x=0 into our expression:
(a⁰1) / 0 = (11) / 0 = 0 / 0. This is undefined and we can use L'Hôpital's rule (check out the link if this is unfamiliar to you). L'Hôpital's rule states that if a limit evaluates to 0/0 (or (±∞)/(±∞), I believe), you can evaluate the limit of the derivative of the nominator divided by the derivative of the denominator (differentiate them both independently). Let's try that:
1. Differentiate both the numerator and the denominator of our expression:
• d/dx [aᵡ  1] = aᵡ · ln(a)  0 = aᵡ · ln(a)
• d/dx [x] = 1
2. Rewrite the original expression as (aᵡ  1)' / x':
(aᵡ · ln(a)) / 1 = aᵡ · ln(a)
3. Reevaluate the limit (let's plug in x=0 once again. If you still end up with 0/0 or (±∞)/(±∞), you can keep on using the rule until you eventually end up with something else):
Plug in x=0:
a⁰ · ln(a) = 1 · ln(a) = ln(a)
We have now proved that the limit of (aᵡ  1) / x = ln(a) as x approaches 0.
An introduction to L'Hôpital's rule:
https://www.khanacademy.org/math/calculus/derivative_applications/lhopital_rule/v/introductiontolhopitalsrule
Could someone please tell me where can I obtain the same emulated graphing calculator that Sal is using? (Please include the download link too!)
To get a virtual calculator, you need to buy a real one and plug it into your computer.
I've heard something about mantessa and characteristic of logarithm...can i find any video on that?
That was a practical part of calculating (base 10) logarithms back before scientific calculators existed. But it isn't something that modern students will ever really need to know, so I wouldn't hold my breath waiting for Sal to make a video about it.
In a nutshell, people used to find logarithms using tables, but the tables would only give the logarithms of numbers between 1 and 9.9999. So, if you wanted to calculate log 5280, you'd say log 5280 = log (1000 * 5.28) = log 1000 + log 5.28 = 3 + log 5.28 = 3.7226. (You'd be able to look up that last bit on the table since it's between 1 and 10.) So you can see how the integer and fractional part of that logarithm are separate keys to finding out what the number is, and the fancy mathematical words for the parts 3 and 0.7226 of that logarithm are the characteristic and mantissa of the logarithm. Of course, nowadays you just push one button on a calculator and it does all that work for you, so the concept doesn't impact much of anyone any more.
In a nutshell, people used to find logarithms using tables, but the tables would only give the logarithms of numbers between 1 and 9.9999. So, if you wanted to calculate log 5280, you'd say log 5280 = log (1000 * 5.28) = log 1000 + log 5.28 = 3 + log 5.28 = 3.7226. (You'd be able to look up that last bit on the table since it's between 1 and 10.) So you can see how the integer and fractional part of that logarithm are separate keys to finding out what the number is, and the fancy mathematical words for the parts 3 and 0.7226 of that logarithm are the characteristic and mantissa of the logarithm. Of course, nowadays you just push one button on a calculator and it does all that work for you, so the concept doesn't impact much of anyone any more.
But in our exams we are not allowed to use scientific calculators..and I can't find any sense in the procedures dictated by my tutor about searching natural sines ,cosines, tangents and all that stuff in a log book...(Clarke's Table)..I really wish Sal made a video..I don't know where else I must post this!
The square root of 69 is 8 something. Right? 'Cause I've been tryin to work it out all....(I still don't get what it means)
(It is a math question, right?)
(It is a math question, right?)
if you use a calculator, it says that sqrt69=8.306623863
Due to Euler's identity does that not mean that the Ln of any algebraic number is transcendental as e^x, if x is algebraic, equals a transcendental number? or did i misunderstand Euler's identity?
Both e^x and ln(x) are transcendental if x is algebraic.
I need help finding a natural log with a variable in the exponent. For example, 5e^(.035x)=200, which reduces to e^(.035x)=40 because I don't know where to go from there.
e^(.035x)=40 From this step, you can manipulate a little bit. So it becomes:
log(base e)40=0.035x
Then go on to solve the problem.
For your example, 0.035x=3.688879(correct to 6 decimal places)
x=105.39656(correct to 5 d.p.)
After you get an answer, try to plug it in. If it is close to the original number, then you have got it right.
Hope this helps :)
log(base e)40=0.035x
Then go on to solve the problem.
For your example, 0.035x=3.688879(correct to 6 decimal places)
x=105.39656(correct to 5 d.p.)
After you get an answer, try to plug it in. If it is close to the original number, then you have got it right.
Hope this helps :)
One thing that I find hard to understand ( altought it may seem a bit silly ) is how can you have a number raise to a noninteger number.... I mean, when you like 2², you multipply 2 times 2. How can you multiply, like, 4.3012312371 times 2? It isn´t really intuitive
```x^(1/n) = nthroot(x)```
I don't really understand the concept of eliminating ln and e together. In my book, an example says:" ln 5x = 4" and there's a step where it converts the problem to 5x = e^4. Can anyone explain it indepth on how the e^ln or ln e = 1 works?
Thanks a lot
Thanks a lot
In my other answer, I demonstrated some of the basic log mixed with exponent properties. Now I will solve the problem you mentioned, but explain each step.
ln 5x = 4
We need to solve for x, so we need to separate it out from either a log or an exponent.
As in demonstrated in my other answer a^(logₐ(x)) = x
Since ln(x) = logₑ(x), ln and e^ undo each other (as shown in my other answer) And,
Using the property that if a=b then nᵃ = nᵇ, and choosing the undo base (e) for the ln x we get:
ln 5x = 4
e^(ln(5x)) = e⁴
Using the undo property of samebase exponent and logarithm:
5x = e⁴
x = ⅕e⁴
Thus x is an irrational and transcendental number. It is approximately:
x ≅ 10.919630
ln 5x = 4
We need to solve for x, so we need to separate it out from either a log or an exponent.
As in demonstrated in my other answer a^(logₐ(x)) = x
Since ln(x) = logₑ(x), ln and e^ undo each other (as shown in my other answer) And,
Using the property that if a=b then nᵃ = nᵇ, and choosing the undo base (e) for the ln x we get:
ln 5x = 4
e^(ln(5x)) = e⁴
Using the undo property of samebase exponent and logarithm:
5x = e⁴
x = ⅕e⁴
Thus x is an irrational and transcendental number. It is approximately:
x ≅ 10.919630
First, understand this:
If a = b, then xᵃ = xᵇ where x is any number.
Now understand that a log and an exponent with the same base undo each other, thus:
logₐ(aˣ) = x
and a^(logₐ(x)) = x
I can go through the proof of this property if it helps, but let me just show you an informal reason:
logₐ(aˣ) = xlogₐ(a)
Since logₐ(n) is asking us what power does a have to be raised to in order to equal n, then if they are the same number, the power is always 1, because a¹=a. Thus, logₐ(a) = 1 (provided a is an allowed number for a logarithm base).
Therefore logₐ(aˣ) = xlogₐ(a) = (x)(1) = x,
Showing that a^(logₐ(x)) = x is a bit more tedious, so I will just show that the equation is true rather than a more direct proof:
a^(logₐ(x)) = x
Take the logₐ of both sides:
logₐ[a^(logₐ(x))] = logₐ(x)
Use the property that log (nᵇ) = (b)log(n)
logₐ(x)(logₐa) = logₐ(x)
Use the property that logₓ(x) = 1 (see above)
logₐ(x)(1) = logₐ(x)
logₐ(x) = logₐ(x)
By law of identity, since both logs have the same base, it must be the case that:
x = x
Thus, the equation is true that a^(logₐ(x)) = x
Since ln(x) means logₑ(x)
Thus ln(eˣ) = x and e^(ln(x)) = x
If a = b, then xᵃ = xᵇ where x is any number.
Now understand that a log and an exponent with the same base undo each other, thus:
logₐ(aˣ) = x
and a^(logₐ(x)) = x
I can go through the proof of this property if it helps, but let me just show you an informal reason:
logₐ(aˣ) = xlogₐ(a)
Since logₐ(n) is asking us what power does a have to be raised to in order to equal n, then if they are the same number, the power is always 1, because a¹=a. Thus, logₐ(a) = 1 (provided a is an allowed number for a logarithm base).
Therefore logₐ(aˣ) = xlogₐ(a) = (x)(1) = x,
Showing that a^(logₐ(x)) = x is a bit more tedious, so I will just show that the equation is true rather than a more direct proof:
a^(logₐ(x)) = x
Take the logₐ of both sides:
logₐ[a^(logₐ(x))] = logₐ(x)
Use the property that log (nᵇ) = (b)log(n)
logₐ(x)(logₐa) = logₐ(x)
Use the property that logₓ(x) = 1 (see above)
logₐ(x)(1) = logₐ(x)
logₐ(x) = logₐ(x)
By law of identity, since both logs have the same base, it must be the case that:
x = x
Thus, the equation is true that a^(logₐ(x)) = x
Since ln(x) means logₑ(x)
Thus ln(eˣ) = x and e^(ln(x)) = x
How would I solve (e^x)^5=1000?
first what exactly is it asking you to solve for e, x, ?
e^5x = 1000 (rule of exponents: power to a power)
=> e^5x/5 = 1000^1/5 (take the 5th root)
e^x = 1000^1/5 (which is approximately 3.98, if rounded to hundredths)
this may get you a little further..
e^5x = 1000 (rule of exponents: power to a power)
=> e^5x/5 = 1000^1/5 (take the 5th root)
e^x = 1000^1/5 (which is approximately 3.98, if rounded to hundredths)
this may get you a little further..
At 5:31,what is mantisa?
Is e an Euler?
Im not understanding the meaning of "e". Where is the derivation or proof shown for e?How come e is 2.71....?
e is a constant that is defined by limits, so if you have not yet studied limits, the definition of e won't mean much to you. Since e is irrational and transcendental, it is impossible to express it exactly by algebraic operations on rational, real numbers.
Here is the definition of e:
e = lim h → 0 (1+h)^(1/h)
Here is the definition of e:
e = lim h → 0 (1+h)^(1/h)
how would you find Ln 3x=2? thank and please show me steps to do it.
how do we simplify expressions such as 2^2x4^3x/16x?
how do i solve the equation to find x 5=ln(x+1)
What does log 3 x mean? Is it like base 3 with power x or you just multiply 3 with x?
Your question is very ambiguous.
`log_3(x) = y` is equivalent to `3^y = x`.
If you were solving for `y` then:
```
3^y = x
ln(3^y) = ln(x)
y•ln(3) = ln(x)
y = [ln(x)/ln(3)]
```
If your expression was instead `log(3^x)` then this simpifies to:
```
[x•log(3)]
```
`log_3(x) = y` is equivalent to `3^y = x`.
If you were solving for `y` then:
```
3^y = x
ln(3^y) = ln(x)
y•ln(3) = ln(x)
y = [ln(x)/ln(3)]
```
If your expression was instead `log(3^x)` then this simpifies to:
```
[x•log(3)]
```
What is e, as a concept? I know it is used for logs and it equals 2.71... but what is the reason for it?
Is there any video that addresses solving or simplifying natural logarithm problems WITHOUT a calculator?
Watch the videos on logarithm properties, and the two video in this play list https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/log_functions/v/solvinglogarithmicequations_DUP_1
what is the first 15 digits of e?
2.71828182845904
ln is latin for logarithm naturalus
What is the exact calculator Sal is using? I know it's a Texas Instruments calculator but I don't know which type it is.
How can i download this calculator that is used in this video........
I am confused when I try to extrapolate this to create an equivalent exponential equation from a logarithmic equation, such as e^2=x+1. Which video should I go to to explain this? Thank you.
why cant you just us log e of n instead of lnx
It's simply because log e is used so much that having a shorter abbreviation makes sense. Mathematically, they are the same. It's just a naming convention.
I do not know what 'e' really is. Is there a video by Sal where he tells what e actually is? In this video he is only telling us how to solve for log_e!
Dont be bothered with the 'e' its a number lyk 'pi'
And its value is some 2.718281828459045 ....already mentioned by just...
And its value is some 2.718281828459045 ....already mentioned by just...
e is an irrational and transcendental number that comes up rather often in math, science, engineering, and even in business math.
Its first few digits are 2.718281828459....
Its first few digits are 2.718281828459....
where and how does e appear in nature
e^0.06t/(1+(0.06/365)^365t = 1000/900
If you use the ChangeofBase formula for log base e of 67:
log base e of 67 = log (67) / log (e)
would you get the exact same answer to ln (67)?
log base e of 67 = log (67) / log (e)
would you get the exact same answer to ln (67)?
Yes Log base e of 67 is the same thing as ln 67
can someone help me figure out how to solve ln(2x1)1=0?
how to use logarithm table
significance of euler number(e)?
what is logarithm
a logarithm is essentially a different way of writing an exponent. It is written like this:
a^x=b
log a (b)=x
a real example of this is:
log 3 (9)=2, because
3^2=9.
each log has a base (in this case, a). in an "ln" or natural logarithm, the base is e. so ln essentially means:
e^x=b
e raised to the xth power makes the number we want.
a^x=b
log a (b)=x
a real example of this is:
log 3 (9)=2, because
3^2=9.
each log has a base (in this case, a). in an "ln" or natural logarithm, the base is e. so ln essentially means:
e^x=b
e raised to the xth power makes the number we want.
At 0:17 Sal said "e" was approx. 2.71..., but wouldn't it be 2.72 because the next digit is 8?
Yes, i think so... e is defined as 2.718281828.
From where can we get the TI85 which Khan Academy uses. If an emulator is there please tell me where i can download it from.
You can probably find it or something like it here: http://www.ticalc.org/programming/emulators/software.html
When do you use natural log and when do you use regular logarithms? I mean, what's the point of using a log with base e when you could just use something simple like log with base 10?
The log is the inverse of the exponential function. So if you had a problem the had say e^x=1 you could simplify this problem by taking the natural log of both sides since it is e to a power and the natural log is log base e.
how do i solve ln(5)1 ?
If you want the exact value, use a calculator.
I don't see anything much you can do with `ln(5)  1`.
I don't see anything much you can do with `ln(5)  1`.
the expression e^x/e^(x+4) can be written as e^f(x), where f(x) is a function of x. find f(x).
can u solve this without a calculator?
if your a super genius and can solve decimal problem with only using your mind
What if e is raised to a power like ee^(2x)=1?
Why is the natural log of a nonpositive number not defined?
Because you can't raise e to any real exponent and get a nonpositive number.
How do you find 3.03=1.086 to the power x without using log?
3.03=1.086^x? I'm afraid you have to use logs for that. Those are the equations when logarithms show their real power :)
If you take log base 1.086 to both sides, you get x= log_1.086 (3.03). Of course, you're not expected to calculate something like that, so if you need a decimal approximation, it's the calculator's job.
If you take log base 1.086 to both sides, you get x= log_1.086 (3.03). Of course, you're not expected to calculate something like that, so if you need a decimal approximation, it's the calculator's job.
How do you express '_*e*_ '?
How do you do natural logs without calculators? Where can I find videos to help me with collegelevel calculus?
so what does 2.h have to do with it all does that help get the answer or do we have to use the calculator
How do you Simplify ln(2e)^2 + ln((4e^6)^0.5)  ln(8e^7) without the use of a calculator?
How long will I t take for $4,000 to grow to $17,000 if the money is invested at 7.7% compounded quarterly?( use the natural logarithm, ln, for this question's solution).
how do you solve
e^e^x = 3
e^e^x = 3
e^eˣ = 3
ln e^eˣ = ln 3
eˣ ln e = ln 3
eˣ = ln 3
ln eˣ = ln(ln 3)
x ln e = ln(ln 3)
x = ln(ln 3)
x ≈ 0.094048
ln e^eˣ = ln 3
eˣ ln e = ln 3
eˣ = ln 3
ln eˣ = ln(ln 3)
x ln e = ln(ln 3)
x = ln(ln 3)
x ≈ 0.094048
I'm still slightly confused  I don't really understand the significance of the number "e." For example, pi is the ratio between the diameter and circumference of a circle, it's used in the Golden Ratio, etc. What's "e" used for, and how did we figure out what its value is?
Thank you both!
Like π, e comes up again and again throughout both pure mathematics and the mathematics of the real world. In fact, in my own profession (Chemistry, though I'm retired now) we run into e far more than we run into π.
e, like π, is both irrational and transcendental, so we cannot write down the exact value of e. It is roughly 2.71828183...
e, like π, is both irrational and transcendental, so we cannot write down the exact value of e. It is roughly 2.71828183...
e is Euler's constant. The number e is of importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity.
Here is a video on Euler's Formula and Euler's Identity that helps explain e.
http://www.khanacademy.org/math/calculus/sequences_series_approx_calc/maclaurin_taylor/v/eulersformulaandeulersidentity
Here is a video on Euler's Formula and Euler's Identity that helps explain e.
http://www.khanacademy.org/math/calculus/sequences_series_approx_calc/maclaurin_taylor/v/eulersformulaandeulersidentity
How did 'e' come into existence? Why use 'e' when you get an approximation?
e is the "natural exponential base". It is typically defined as the limit of: (1+1/n)^n as n approaches infinity (though I'll agree that isn't the most intuitive definition). Alternatively, the rate of increase of e^x at any point on the graph is equal to the value of the function itself. e^x defines continuous exponential growth at a 100% growth rate.
In calculus, the derivative of ln(x) = 1/x and the derivative of e^x = e^x. While you can also take the derivative of functions using other bases, these formulas are "clean" in that they don't need any additional constants to adjust. And using a Taylor series, e^x = 1 + x + x^2/2 + x^3/6 + .... + x^n/n! + .... Again, a "clean" formula.
It is a little like pi with circles. If a circle has an integer radius, then the circumference and area will be most easily (and exactly) expressed as a multiple of pi. Similarly, continuous exponential growth is most easily and exactly expressed using a base of e.
In calculus, the derivative of ln(x) = 1/x and the derivative of e^x = e^x. While you can also take the derivative of functions using other bases, these formulas are "clean" in that they don't need any additional constants to adjust. And using a Taylor series, e^x = 1 + x + x^2/2 + x^3/6 + .... + x^n/n! + .... Again, a "clean" formula.
It is a little like pi with circles. If a circle has an integer radius, then the circumference and area will be most easily (and exactly) expressed as a multiple of pi. Similarly, continuous exponential growth is most easily and exactly expressed using a base of e.
He said "just as a reminder". This implies he talked about e before. I can't find it in the playlists before though. Where did he introduce it?
I'm not sure where it was introduced but here's some basic videos on it:
https://www.khanacademy.org/math/algebra2/exponential_and_logarithmic_func/continuous_compounding/v/introductiontointerest
https://www.khanacademy.org/math/algebra2/exponential_and_logarithmic_func/continuous_compounding/v/introductiontointerest
To be Ln(xy)<0, the solution was e is greater than one so xy is less one. Can you explain me why? Thank you!
how do you graph log base 2 of (x + 4) on the ti83 plus calculator