<< Chapter < Page Chapter >> Page >
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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluate the following integral: 4 e 3 x d x .

4 e 3 x d x = 4 3 e −3 x + C

Got questions? Get instant answers now!

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.

Questions & Answers

A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
anyone to explain some basic in calculus
Adegoke Reply
I can
Debdoot

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask