Chapter 5
Exponential and
Logarithmic
Functions
5.1 Exponential Functions
Exponential Functions
For b > 0, b≠1, f(x) = bx defines the
base b exponential function.
The domain of f is all real numbers.
5.1 Exponential Functions
Exponential Properties
Given a, b, x, and t are real numbers, with b, c > 0,
b b b
x t
bc 
x
x t
b c
x
x
x
b
x t

b
t
b
b
x
1
 x
b
b   b
x t
b
 
a
x
xt
a
 
b
x
5.1 Exponential Functions
Graphs of exponential functions
Important Characteristics
One-to-one function
Domain: x  R
Y-intercept (0,1)
Range: y  0, 
5.1 Exponential Functions
f x   b , b  0 and b  1
x
Increasing if b>1
Decreasing if 0<b<1
5.1 Exponential Functions
EXPONENTIAL EQUATIONS WITH LIKE BASES
THE UNIQUENESS PROPERTY
If bm = bn,
then m = n.
If m = n,
then bm = bn.
2 x 1
3
 81
2 x 1
4
3
3
2 x 1  4
5
x
2
5.1 Exponential Functions
EXPONENTIAL EQUATIONS WITH LIKE BASES
THE UNIQUENESS PROPERTY
25  125
3x
5 
2 3x
x2
 
3 x2
 5
3 x 6
5 5
6x
6x  3x  6
x  2
5.1 Exponential Functions
Homework pg 482 1-68
5.2 Logarithms and Logarithmic Functions
Logarithmic Functions
For b > 0, b ≠ 1, the base-b logarithmic function is defined as
y  log b x if and only if x  b
Write in exponential form
3  log 2 8
0  log 2 1
23  8
20  1
y
Write in logarithmic form
1
2 
2
1
1
 1  log 2
2
3
2
9  27
3
 log 9 27
2
5.2 Logarithms and Logarithmic Functions
Graphing Logarithmic Functions
Calculators and Common Logarithms
5.2 Logarithms and Logarithmic Functions
Pg 493 #87 and 88
Earthquake Intensity
5.2 Logarithms and Logarithmic Functions
Homework pg 491 1-94
5.3 The Exponential Function and Natural Logarithms
Natural Logarithmic Function
5.3 The Exponential Function and Natural Logarithms
Properties of Logarithms
Given M, N, and b are positive real numbers, where
b ≠ 1, and any real number x.
Product Property: log b
MN   log b M  log b N
“the log of a product is equal to a sum of logarithms”
Quotient Property: log b
M
 log b M  log b N
N
“The log of a quotient is equal to a difference of logarithms”
Power Property:
x


log b M  x  log b M
“The log of a number to a power is equal to the power times
the log of the number”
5.3 The Exponential Function and Natural Logarithms
Using Properties of Logarithms
 m2 
ln  3 
n 

log x 4 y

5.3 The Exponential Function and Natural Logarithms
Using Properties of Logarithms
log 3 28  log 3 7


log 5 x 2  2 x  log 5 x 1
5.3 The Exponential Function and Natural Logarithms
Change of Base Formula
Given the positive real numbers M, b, and d, where b≠1 and d≠1,
log M
log b M 
log b
base 10
ln M
log b M 
ln b
base e
5.3 The Exponential Function and Natural Logarithms
Using the change of base formula
log 5 152
log 0.2 0.008
5.3 The Exponential Function and Natural Logarithms
Homework pg 502 1-106
5.4 Exponential/Logarithmic Equations and Applications
Writing Logarithmic and Exponential Equations in Simplified Form
log 2 x  log 2 x  3  4
log 2 xx  3  4


log 2 x  3x  4
2
 ln 2 x  ln x  ln x  1
x
 ln 2 x  ln
x 1
x
0  ln
 ln 2 x
x 1
 x  2 x 
0  ln 
 
 x  1  1 
 2x2 

0  ln 
 x 1
5.4 Exponential/Logarithmic Equations and Applications
Writing Logarithmic and Exponential Equations in Simplified Form
400e
0.21x
 325  1225
400e 0.21x  900
e
0.21x
 2.25
e
x 1
e   e
3x
2x
e 4 x 1  e 2 x
4 x 1
e
1
2x
e
e 4 x 1 2 x  1
e
2 x 1
1
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations
For any real numbers b, x, and k, where b>0 and b≠1
if 10 x  k ,
if e  k ,
log 10 10  log 10 k
ln e  ln k
 x  ln k
x
 x  log 10 k
x
x
if b x  k ,
x log b  log k
log k
x
log b
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations
3e
x 1
3e
e
ln e
5  7
x 1
 12
x 1
4
x 1
 ln 4
x  1  ln 4
x  ln 4 1
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations
258
 192
 0.009t
1  20e

258  192 1  20e 0.009t
258
 1  20e 0.009t
192
258
 1  20e 0.009t
192
258
1
192
 e 0.009t
20

 258 
1 

ln  192   ln e 0.009t
 20 




 258 
1 

ln  192   0.009t
 20 




 258 
1 

ln  192 
 20 



 t
 0.009
5.4 Exponential/Logarithmic Equations and Applications
Solving Logarithmic Equations
For real numbers b, m, and n where b > 0 and b≠1,
if log b m  log b n
then m  n
if m  n
then log b m  log b n
Equal bases imply equal arguments
5.4 Exponential/Logarithmic Equations and Applications
Solving Logarithmic Equations
log x  12  log x  log x  9
 x  12 
log 
  log  x  9 
 x 
x  12
 x9
x
x  12  x 2  9 x
0  x 2  8 x  12
Use quadratic formula to solve for x
5.4 Exponential/Logarithmic Equations and Applications
An advertising agency determines the number of items sold is
related to the amount spent on advertising by the equation
N(A)= 1500 + 315 ln A, where A represents the advertising
budget and N(A) gives the number of sales. If a company wants
to generate 5000 sales, how much money should be set aside for
advertising? Round interest to the nearest dollar.
N  A  1500  315 ln A
5000  1500  315 ln A
3500  315 ln A
3500
 ln A
315
e
3500
315
e
ln A
e
3500
315
A
5.4 Exponential/Logarithmic Equations and Applications
Homework pg 516 1-106
Chapter 5 Review