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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 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.
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.

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.