Warm-Up
Solve for x in each equation.
a) 3x =
c) 5x = 30
b) log2x = –4
5
Logarithmic, Exponential, and Other
Transcendental Functions 2014
If you aren't in over
your head, how do
you know how tall
you are?
T. S. Eliot
Copyright © Cengage Learning. All rights reserved.
Chapter 5
Transcendental Functions:
Bases Other than e
Day 1: All rules and derivative examples.
Day 2: Integration examples and applications.

http://www.youtube.com/watch?v=SNZgbj3Ua
RE
Bases Other than e

Recall that the natural log, ln x, is a logarithm
with base e. We can differentiate ln x simply with
the following rule:
d
ln x   1
dx
x

In addition, we know that the natural exponential
function, ex, can easily be differentiated:
 
d x
e  ex
dx

By applying the chain rule, we get:
d
u
ln u  
dx
u
and
 
d u
e  eu  u
dx
Bases Other than e

But, what do we do if we have a logarithm or
exponential function that has a base other than e?
To determine these derivatives, we need to use a
very useful logarithmic operation called the
change of base formula. Recall from precalculus:
loga b 

logc b
logc a
This formula allows us to transform any
logarithm into a quotient of logarithms with any
base that we choose, including the natural
logarithm. More specifically:
loga b 
loge b ln b

loge a ln a
Bases Other than e

Now let’s try to find the derivative of a logarithm
with a base other than e:
d  ln x 
d
log a x      1  d ln x  1  1
dx  ln a  ln a dx
dx
ln a x
This is a constant
Bases Other than e

To determine the derivative of a natural
exponential function with a base other than e, we
need to note: a x  elna  e xlna
x
 
d x
a
dx

d xln a

e
dx

 

u  x  ln a
u   ln a
d u
e  eu  u  e x ln a  ln a  ln a  a x
dx
Bases Other than e

In terms of integration, at this point, we cannot
find the integral of ln x, but we can integrate ex.
Recall:
e x dx  e x  C


So, how do we integrate an exponential function
with a base other than e? Once again, we use the
alternate form of the exponential function of ax:
a e
x
ln a x
x
x ln a
a
dx

e

 dx u  x  ln a
1
u

e
du

ln a
 e xlna
d u   ln a  d x
1 u
1 x ln a

e C 
e
C
ln a
ln a
1 x

a C
ln a
Rules for derivatives of bases Other than e

The rules to differentiating with bases other than e
d
1 1
1. log a x  

dx
ln a x
3.

 
d x
a  ln a  a x
dx
d
1 u
log a u   
2.
dx
ln a u
4.
 
d u
a  ln a  a u  u 
dx
The rules to integrating with bases other than e
1 x
5.  a dx 
a C
ln a
x
6.  a u du 
1 u
a C
ln a

d
log a u   1  u
dx
ln a u
Examples

Differentiate:

1. y  log7 x 2  5x

 
d u
a  ln a  a u  u 
dx
1 2x  5
2x  5
y 
 2
ln 7 x  5 x (ln 7 )( x 2  5 x )


2. y  log4 x 2  cos x  lo g 4 x 2  lo g 4 c o s x  2 lo g 4 x  lo g 4 c o s x
1 1
1  sin x
2
tan x
2
1
y  2 
 




 tan x 
ln 4 x ln 4 cos x
x ln 4 ln 4
x ln 4 ln 4

3. y  7
x3  x 2 1
y   ln 7  7  u   ln 7  7
u
4. y  x3 8x
du
 3x2  2x
dx
u  x  x 1
3
2
x 3  x 2 1
1 2
  tan
ln 4  x
( 3 x  2 x )  (3 x  2 x )(7
2
y   3 x 2 8 x  x 3 8 x ln 8
2
x 3  x 2 1

x

) ln 7
Practice Problems

d
1 u
log a u   
dx
ln a u
Differentiate:

1. y  log2 2 x  4
5

3
1
10 x 4
y  3 
 5
ln 2 2 x  4


 3 log2 2x5  4
30 x 4

ln 2 ( 2 x 5  4 )
u  2x5  4
15 x 4

ln 2 ( x 5  2 )
u  10 x4
Practice Problems

d
1 u
log a u   
dx
ln a u
Differentiate:
x2
2. y  log 7 2
x
y 
 log7 x  2  log7 x2
1
1
1 1

 2

ln 7 x  2
ln 7 x
 log7 x  2  2 log7 x
1
2


(ln 7 )( x  2 ) (ln 7 ) x
x
2x  4
x4
4x




x (ln 7 )( x  2 ) x (ln 7 )( x  2 ) x (ln 7 )( x  2 )
(ln 7 )( x 2  2 x )
Practice Problems

Differentiate:
1. y   x
2. y  10
x2 sin x
d x
 a   ln a  a x
dx
 ln   
x
 
d u
a  ln a  a u  u 
dx
  x ln 
u  x 2  s in x
 ln 1 0  1 0
u   2 x  cos x
x 2  sin x
2 x  c o s x 
  2 x  c o s x 1 0
x 2  sin x
ln 1 0
4
x
x
5
4 x
5 x
3. y  x5 9x  5 x  9  ln 9  9  x  5 x 9  x 9 ln 9
5.5 Homework Day 1 AB
Page 366
 1, 7, 19, 21, 27, 37-51 odd, 57

5.5 Homework Day 1 BC
Page 366
 21-55 odds

Chapter 5
Transcendental Functions:
Bases Other than e
Day 2: Integration examples and Applications.
18
HWQ

Find an equation of the tangent line to the graph
of y  log10 2x at 5,1 .
1
y 1 
 x  5
5 ln10
Day 2: Integration
Occasionally, an integrand involves an exponential
function to a base other than e. When this occurs,
there are two options:
(1) use substitution, and then integrate, or
(2) integrate directly, using the integration formula
Example – Integrating an Exponential Function to Another Base
Find ∫2xdx.
Solution:
∫2xdx =
+C
Examples

Integrate:
5.  37 x dx
u  7x
1 u
  3 du
7
6.  x5 dx
x2
u  x2
du  7 dx
1 1
 
 3u  C
7 ln 3
37 x

C
7 ln 3
du  2 x dx
1 u
1 1
  5 du  
 5u  C
2
2 ln 5
x2
5

C
2 ln 5
Practice Problems

Integrate:
7.  5 6 x dx
u  6x
du  6 dx
6x
5
1 1
1 u
C
 5u  C 
  5 du  
6 ln 5
6 ln 5
6
8.  (3  x)7
( 3 x ) 2
dx
1 u
   7 du
2
u  (3  x ) 2
d u   2 3  x d x
1 1
 
 7 (3 x )  C
2 ln 7
2
(3 x ) 2
7

C
2 ln 7
Applications of Exponential
Functions
Applications of Exponential Functions
Suppose P dollars is deposited in an account at an annual
interest rate r (in decimal form). If interest accumulates in
the account, what is the balance in the account at the end of
1 year? The answer depends on the number of times n the
interest is compounded according to the formula
A=P
Applications of Exponential Functions
For instance, the result for a deposit of $1000 at 8%
interest compounded n times a year is shown in
the table.
Applications of Exponential Functions
As n increases, the balance A approaches a limit. To
develop this limit, use the following theorem.
Applications of Exponential Functions
To test the reasonableness of this theorem, try evaluating
it for several values of x, as shown in the table.
Applications of Exponential Functions
Now, let’s take another look at the formula for the balance
A in an account in which the interest is compounded n
times per year. By taking the limit as n approaches infinity,
you obtain
Applications of Exponential Functions
This limit produces the balance after 1 year of continuous
compounding. So, for a deposit of $1000 at 8% interest
compounded continuously, the balance at the end of 1 year
would be
A = 1000e0.08
≈ $1083.29.
Applications of Exponential Functions
Example 6 – Comparing Continuous, Quarterly, and Monthly Compounding
A deposit of $2500 is made in an account that pays an
annual interest rate of 5%. Find the balance in the account
at the end of 5 years if the interest is compounded
(a) quarterly, (b) monthly, and (c) continuously.
Solution:
Example 6 – Solution
cont’d
Example 6 – Solution
cont’d
Figure 5.26 shows how the balance increases over the fiveyear period. Notice that the scale used in the figure does not
graphically distinguish among the three types of exponential
growth in (a), (b), and (c).
Figure 5.26
5.5 Homework Day 2
Page 366
 59-71 odds, 83, 85
