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exp ( ln x ) = x for x > 0 and ln ( exp x ) = x for all x .

The following figure shows the graphs of exp x and ln x .

This figure is a graph. It has three curves. The first curve is labeled exp x. It is an increasing curve with the x-axis as a horizontal asymptote. It intersects the y-axis at y=1. The second curve is a diagonal line through the origin. The third curve is labeled lnx. It is an increasing curve with the y-axis as an vertical axis. It intersects the x-axis at x=1.
The graphs of ln x and exp x .

We hypothesize that exp x = e x . For rational values of x , this is easy to show. If x is rational, then we have ln ( e x ) = x ln e = x . Thus, when x is rational, e x = exp x . For irrational values of x , we simply define e x as the inverse function of ln x .

Definition

For any real number x , define y = e x to be the number for which

ln y = ln ( e x ) = x .

Then we have e x = exp ( x ) for all x , and thus

e ln x = x for x > 0 and ln ( e x ) = x

for all x .

Properties of the exponential function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of e , we must verify that the usual laws of exponents hold for the function e x .

Properties of the exponential function

If p and q are any real numbers and r is a rational number, then

  1. e p e q = e p + q
  2. e p e q = e p q
  3. ( e p ) r = e p r

Proof

Note that if p and q are rational, the properties hold. However, if p or q are irrational, we must apply the inverse function definition of e x and verify the properties. Only the first property is verified here; the other two are left to you. We have

ln ( e p e q ) = ln ( e p ) + ln ( e q ) = p + q = ln ( e p + q ) .

Since ln x is one-to-one, then

e p e q = e p + q .

As with part iv. of the logarithm properties, we can extend property iii. to irrational values of r , and we do so by the end of the section.

We also want to verify the differentiation formula for the function y = e x . To do this, we need to use implicit differentiation. Let y = e x . Then

ln y = x d d x ln y = d d x x 1 y d y d x = 1 d y d x = y .

Thus, we see

d d x e x = e x

as desired, which leads immediately to the integration formula

e x d x = e x + C .

We apply these formulas in the following examples.

Using properties of exponential functions

Evaluate the following derivatives:

  1. d d t e 3 t e t 2
  2. d d x e 3 x 2

We apply the chain rule as necessary.

  1. d d t e 3 t e t 2 = d d t e 3 t + t 2 = e 3 t + t 2 ( 3 + 2 t )
  2. d d x e 3 x 2 = e 3 x 2 6 x
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Evaluate the following derivatives:

  1. d d x ( e x 2 e 5 x )
  2. d d t ( e 2 t ) 3
  1. d d x ( e x 2 e 5 x ) = e x 2 5 x ( 2 x 5 )
  2. d d t ( e 2 t ) 3 = 6 e 6 t
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Using properties of exponential functions

Evaluate the following integral: 2 x e x 2 d x .

Using u -substitution, let u = x 2 . Then d u = −2 x d x , and we have

2 x e x 2 d x = e u d u = e u + C = e x 2 + C .
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Evaluate the following integral: 4 e 3 x d x .

4 e 3 x d x = 4 3 e −3 x + C

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General logarithmic and exponential functions

We close this section by looking at exponential functions and logarithms with bases other than e . Exponential functions are functions of the form f ( x ) = a x . Note that unless a = e , we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function f ( x ) = a x in terms of the exponential function e x . We then examine logarithms with bases other than e as inverse functions of exponential functions.

Definition

For any a > 0 , and for any real number x , define y = a x as follows:

y = a x = e x ln a .

Now a x is defined rigorously for all values of x . This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of r . It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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