5.3
Exponential and Logarithmic Functions
Logarithms
Recall that
y  log b x
 by  x
Logarithmic form
Example:
Exponential form
log 2 32
Simplify:

x  0,b  0,b  1
log 4 32

Solve: 5
4 x7
 3  10

Properties of Logarithms ( b  0,b  1)
log b 1 
log b b 
log b b x 
b logb x 
log b (mn) 
m 
log b  
n 
log1 

log10 
log10 x 
10 logx 
ln1 
ln e 
ln e x 
e ln x 
log b m p 
1

Example:
Given ln2 = 0.69315 and ln10 = 2.30259, simplify:
ln 2000
Example:
ln 0.002
Given a = ln3 and b = ln5, simplify:
ln 27
ln 3 25
ln 45
ln e(x 1)
e(x 2 ln x )

Example:
Simplify:
e 3 ln x

2
Inverse Functions
A __________________ function is a function in which no two different ordered
pairs have the same second component. An easy way to visually test the graph is to use
the __________________ __________ test. This test insures that the graph’s inverse is a
function.
Example:
Is each function one-to-one? Sketch to test.
F(x)  2x 2  5
G(x)  x  7

Two Ways to Determine the Existence of an Inverse Function:
1) Horizontal Line Test
2) Use the first derivative to see if the function is strictly monotonic.
Steps for Finding the Equation for an Inverse Function:
Step 1:
Step 2:
Step 3:
Step 4:
Remember*
*The graphs of inverse functions are symmetric to the line _________.
*The domain of f 1 is the ____________ of f.
*You can verify inverses algebraically by showing that:

3
Example:
Determine if the function has an inverse function. If so, (a) state the
domain of f and f 1 , graph both of the functions.
f (x)  x 3  x 1

f (x)  x 3  x  1
f (x)  x  1

4
Derivative of an Inverse Function
Exploration:
Graph the inverse functions
1
f (x)  x 3

and g(x)  x 3
Calculate the slope of f at (1,1), (2,8), and (3,27), and the slope of g
at (1,1), (8,2), (27,3). What do you observe? What happens at
(0,0)?
See if you can describe what you observed in a more formal way:
The Derivative of an Inverse Function
Let f be a function that is differentiable on an interval I. If f has an inverse function g,
then g is differentiable at any x for which f '(g(x))  0 . Moreover,
g'(x) 


We can actually prove this using the Chain Rule:
5
Example:
f (x) 
1 3
x  x 1
4
Find ( f 1 )'(3).

Example:
Find ( f 1 )'(a) for each function.
f (x)  sin x,

f (x)  x  4


2
x

2
a
1
2
a2

6
5.1,5.4,5.5
Derivatives of Exponential and Logarithmic Function
Derivative of the Natural Exponential Function
Graph y  e x
and
y
d x
[e ]
dx
What do you notice?


d x
[e ] 
dx

Examples:
d 2x1
[e ] 
dx
3
d x
[e ] 
dx

7
Derivative of the Natural Logarithmic Function
and y  ln x are inverse functions. We can use that
We know that y  e x
relationship to derive the derivative of ln x:


d
[ln x] 
dx
Important to Note:

x 0
d
[ln x ] 
dx
for x  0
Examples: Find the derivative of each:

f (x)  ln( x 2  3)
g(x)  ln x 2  4
y  ln(ln x)
d  1 e x 
ln 

dx  1 e x 
y  ln 2x  3

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Derivatives of Bases Other Than e
Consider
log a x 
a x  e(ln a )(x )
d x
d
[a ]  [e(ln a )(x ) ]
dx
dx


d
d  1
[log a x]    ln x

dx
dx ln a



ln x
ln a
d x
[a ] 
dx
Examples:

d
[log a x]
dx

Find the derivative of each of the following:
y  2x
y  2 3x
y  log 7 (2x)
x 2 1
y  log 3  3 
 x 

9
Domain, Zeros, and Y-Intercept of Exponential and Logarithmic Functions
For each of the following, state the
a) domain
b) the zeros
c) y-intercept
y  ex
y  log a (x)
y  ln x
f (x)  x log 2 (3x  1)
[ln( x  1)][e x 1 ][ 2x  3]
f (x) 
5x  7
2

10
Using the First Derivative Test for Exponential and Logarithmic Function
For each of the following, find the
a) domain
b) increasing/decreasing intervals
c) relative extrema
F(x)  2 x
2
1
G(x)  2
x2
1
H(x)  2 x
2
9
J(x)  log x 2  9

11
7.7
Indeterminate Forms and L’Hopital’s Rule
Recall when evaluating limits that 0/0 and / are called indeterminate, because
they do not guarantee that a limit exists, nor do they indicate that the limit does not exist.
One way that you can try to evaluate these limits is by rewriting the quotient:

0 lim 2x 2  2

0 x  1 x  1
 lim 3x 2 1

 x   2x 2  1
Not all indeterminate forms can be evaluated by algebraic manipulation.


lim e 2x 1
x 0 x
Evaluating the limit numerically or graphically shows that the limit exists, but may not
allow you to determine the exact answer. We can evaluate this limit analytically by using
L’Hopital’s Rule.
L’Hopital’s Rule
Let f and g be functions that are differentiable on an open interval (a,b) containing c,
except possibly at c itself. Assume that g'(x)  0 for all x in (a,b), except possibly at c
itself. If the limit of f(x)/g(x) as x approaches c produces an indeterminate form, then

Example:
lim e 2x 1
x 0 x
Evaluate:
lim ln x
x  x
lim x 2
x   ex
(L’Hopital’s Rule can be applied more than one.)

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Other indeterminate forms: 0  ,
1 ,  0, 0 0 ,

We can use L’Hopital’s Rule to evaluate a very important limit:

lim 
1
x   
1 

x 
x

13