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# How does e^x equal a natural logarithm?

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.

Thank you Steve, Tom, and Kaili! You guys rock!

### 3 Answers by Expert Tutors

Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
5.0 5.0 (3 lesson ratings) (3)
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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.

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)
Kaili S. | Algebra I, II, College Algebra, Geometry, Certified teacher 7-12Algebra I, II, College Algebra, Geometry...
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Alright. So you know what logax=y means, right? If not, a little review here: a, is called the base; y, is called the exponent. If we have ay=x, we would automatically have logax=y as well. So for instance, we could have 42=16, if we reverse that, we would get, log416=2. Notice that, usually, the base would stay the same, but x and y changes accordingly. So in this case of log4x=y, I could also have log464=3, (because 43=64) …etc.

Now, lnx, is a special case of logax: when the base a is "e". When the base is e, we write logex as lnx.
So, if I have e1=e, I then will have logee=lne=1.
e2=p (p is just a value, too long to type out), I then will have logee2=lne2=lnp=2
ex=q (q is also just a value, too long to type out), I then will have logeex=lnex=lnq=x
Hope it helps.
Tom D. | Very patient Math Expert who likes to teachVery patient Math Expert who likes to te...
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ln(ex) = 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:

ei*pi+1 = 0

or more generally:

e=cos(Θ)+isin(Θ)  <---Eulers equation

e crops up in much of math & physics.  It is simply ubiquitous.