I'm learning about this in class. The teacher switches ln for e^x and the numbers all switch. I'm just confused as to why this works.

## How does e^x equal a natural logarithm?

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# 3 Answers

Ben,

Here's another perspective:

You know how to graph y = b^x, a simple exponential function.

You also know how to find the inverse of a function by reversing the x and y (this reflects the graph of the function over the line y = x).

If you try to find the inverse of the simple exponential function you get x = b^y. It's easy to graph this function: you just take all the points you used to graph y = b^x and swap the coordinates, graph those and draw a smooth curve through them. If you
do it you'll see the graph of the inverse is a reflection over y=x of the exponential function.

But the problem is, how do you solve for y in the inverse function x = b^y? There is no way you can do it without defining a new function name; that new function name is the "logarithm". The logarithm function has the same information in it as does x =
b^y, i.e., y = log_b(x). See the x, y, and b in both expressions? (The b is printed as a subscript usually, I'm using "_b" to indicate subscript.)

You may be given one of these expressions and asked for the equivalent other one; just identify the three numbers in the given expression and put them in the right place in the other.

There are two shorthand notations when the base is 10 or e: log and ln. You will see these on calculators.

Finally, please always use parenthesis to enclose the input to any function, including logs. E.g., log_3(27), not log_327.

# Comments

Steve makes a great point. Plot the functions e^x & ln(x) in a google browser to visualize what he is saying.

Just type the two expressions separated by a comma (Use '^' for the exponent)

Alright. So you know what

**log**means, right? If not, a little review here:_{a}x=y**a**, is called**the base**;**y**, is called the**exponent**. If we have**a**we would automatically have^{y}=x,**log**as well. So for instance, we could have 4_{a}x=y^{2}=16, if we reverse that, we would get, log_{4}16=2. Notice that, usually, the base would stay the same, but x and y changes accordingly. So in this case of log_{4}x=y, I could also have log_{4}64=3, (because 4^{3}=64) …etc.Now,

**lnx**, is a special case of**log**_{a}x: when the base a is "e". When the base is e, we write log_{e}x as lnx.So, if I have e

^{1}=e, I then will have log_{e}e=lne=1. e

^{2}=p (p is just a value, too long to type out), I then will have log_{e}e^{2}=lne^{2}=lnp=2…

e

^{x}=q (q is also just a value, too long to type out), I then will have log_{e}e^{x}=lne^{x}=lnq=xHope it helps.

ln(e

^{x}) = x By definition where the base is 'e' and ln() is the 'natural logarithm' Typically 'log()' is reserved for base=10 logarithms.e=∑(1/n!) (n=0-->infinity) = lim(n-->inf) of (1+1/n)

^{n}'e' is the most fascinating number that exists (IMHO). Even pi takes 2nd place to 'e' Eulers' identity is also perhaps the most incredible relationship in all of math:

e

^{i*pi}+1 = 0or more generally:

e

^{iΘ}=cos(Θ)+isin(Θ) <---Eulers equatione crops up in much of math & physics. It is simply ubiquitous.

# Comments

Ben,

Repeat after me 20 times:

**"A Logarithm is the exponent of the base.**" Say it while you are going to sleep and as you wake and they mystery will melt away.
## Comments

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