How does e^x equal a natural logarithm?

Question

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. 

Answer 1

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.

Answer 2

Alright. So you know what log_ax=y means, right? If not, a little review here: a, is called the base; y, is called the exponent. If we have a^y=x, we would automatically have log_ax=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 log_ax: when the base a is "e". When the base is e, we write log_ex as lnx.

So, if I have e¹=e, I then will have log_ee=lne=1.

                   e²=p (p is just a value, too long to type out), I then will have log_ee²=lne²=lnp=2

…

                   e^x=q (q is also just a value, too long to type out), I then will have log_ee^x=lne^x=lnq=x

Hope it helps.

Answer 3

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 = 0

 

or more generally:  

 

e^iΘ=cos(Θ)+isin(Θ)  <---Eulers equation

 

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