Lehigh University Math 21 - Zoo of Functions: Basic Definitions

Derivatives of e^x and ln(x)

Limits of exponential functions

Exponential functions always seem to involve this mysterious number e . It's definition is wrapped up with limits and derivatives of exponential functions. So, what's e ? The number e is chosen to be that one number so that the exponential function with that as its base passes through (0,1) with slope 1. Recall the picture of the various exponential graphs:

That's really the primary reason why e is chosen as it is. It's similar to the reason we use radian measure for trigonometric functions: things work out more conveniently that way.

To see what makes the slope at (0,1) so important, look at the derivation of the derivative of f(x) = a^x , for an arbitrary a :

f¢(x) = ( a^x) ¢

: =

lim
h® 0
a^(x+h)-a^x
h

=

lim
h® 0
a^xa^h-a^x
h

=

lim
h® 0
a^x( a^h-1)
h

=

a^x
lim
h® 0
( a^h-1)
h
.

The last part,

lim
h® 0
(a^h-1)/h,

is the slope of the curve y = a^x at (0,1). If we can choose a so that that last bit is 1, we really have simplified the situation. So, define e to be that base for which the slope is 1, and you have a neater formula. Does there exist such a base? Certainly you can find one with slope 0, or negative, and you can find one with huge slope at (0,1), like y = 1000^x , so somewhere in between you can find one with slope 1. (This, by the way, uses the Intermediate Value Theorem, which will be dealt with officially in the next chapter). That base is e , and its value is 2.71828... Here is a picture of just y = 2^x, y = 3^x, and y = x+1 . The last one is a line with slope 1 going through (0,1). You should notice that y = 2^x does not have enough slope at (0,1), while y = 3^x has too much.

Look back again at that derivation of the derivative of a^x . It shows that:

( e^x) ¢ = e^x,

which seems odd, but is true. The slope at each value of x is the value at that x . This property is important in using the exponential function as a model of systems, but for now, that's just the derivative formula. From now on, though, f(x) = e^x is just another function to add to our list. You use all the other derivative rules with this function like you any other function.

3.3.2 Logarithms.

Logarithms are just exponents in reverse:

y = log_bx Û x = b^y.

So, log₄2 = 1/2 , log₃(1/3) = -1 , and so on. There is a special case, when the base is e . Then, instead of writing log_ex , we write lnx , the natural logarithm of x . There is also another notation for log₁₀x . It is sometimes called logx , the common logarithm of x . Unfortunately, most post-calculus math texts now confuse the notational issue even more, by writing the natural logarithm of x as logx . Hopefully, it will be clear from the context which is meant. To a mathematician, there is no special significance of the number 10, except for the number of fingers we have.

The derivative of the natural logarithm is figured out by the using the fact that the logarithm is the inverse of the exponential. Really, though, it's just the chain rule: if f(x) = lnx , then

e^lnx

=

x, and so

( e^lnx) ¢

=

(x)¢

e^lnx(lnx)¢

=

1

x (lnx)¢

=

1, or

(lnx)¢

=

1
x
.

The text has a couple of other formulas that are really the same as this one, but fancier-looking since they stick the chain rule in there:

( ln(g(x)) ¢ = g¢(x)
g(x)
,

and

( ln|x|) ¢ = 1
x
.

This last one looks like the same formula, except with the absolute value sign. But remember that lnx itself is defined only for x > 0 , so this is a way to extend it to x < 0 - sort of. For x < 0 , set u = -x , then, using the version with the chain rule:

( ln|x|) ¢

=

( ln(u)) ¢

=

u¢
u

=

-1
|x|

=

1
x
, since x < 0.

So, the formula does work the same for either positive or negative values of x .