Chapter 4 Study Guide
4.1 Exponential Functions, Growth, and Decay
The base of an exponential function indicates whether the function shows growth or decay.
Exponential function: f (x)  ab x
• a is a constant.
• b is the base. The base is a constant.
If 0  b  1, the function shows decay.
If b  1, the function shows growth.
• x is an exponent.
When an initial amount, a, increases or decreases by a constant rate, r, over a number of time periods, t, this
formula shows the final amount, A (t).
A (t)  a (1  r)t
An initial amount of $15,000 increases by 12% per year. In how many years will the amount reach $25,000?
Step 1
Identify values for a and r.
a  $15,000
Step 2
r  12  0.12
Substitute values for a and r into the formula.
f (t)  a (1  r)t
Remember:
r the
12%
 0.12
On
graph,
x
corresponds to t and y
corresponds to f (t).
f (t)  15,000 (1  0.12 )t
f (t)  15,000 (1.12)t
Simplify.
Step 3
Graph the function using a graphing calculator.
Modify the scales: [ 0, 10 ] and [ 0, 30,000 ].
Step 4
Use the graph and the [TRACE] feature
on the calculator to find f (t)  25,000.
Step 5
Use the graph to approximate the value of t
when f (t)  25,000.
t  4.5 when f (t)  25,000
The amount will reach $25,000 in about 4.5 years.
Tell whether each function shows growth or decay. Then graph.
1. h(x)  0.8(1.6)x
2. p(x)  12(0.7)x
Write an exponential function and graph the function to solve.
3. An initial amount of $40,000 increases by 8% per year. In how many years
will the amount reach $60,000?
a. f (t)  ______________________
b. Approximate t when f (t)  60,000 t  ____________
4.2 Inverses of Relations and Functions
To graph an inverse relation, reflect each point across the line yx.
Or you can switch the x- and y-values in each ordered pair of the relation to find the ordered pairs of the
inverse.
Inverse operations undo each other, like addition and subtraction, or multiplication
and division.
In a similar way, inverse functions undo each other.
The inverse of a function f (x) is denoted f1(x).
Use inverse operations to write inverse functions.
Function: f(x)  x  8
Subtraction is the opposite of addition.
Use subtraction to write the inverse.
Function: f(x)  5x
Division is the opposite of multiplication.
Use division to write the inverse.
f 1( x ) 
Inverse: f1(x)  x  8
Choose a value for x to check in the
original function. Try x  1.
f(x)  x  8  f(1)  1  8  9
Substitute 9, into f1 (x). The output of
the inverse should be 1.
f1 (x)  x  8  f 1 (9)  9  8  1
Think: (1, 9) in the original function should
be (9, 1) in the inverse. 
Inverse:
x
5
Choose a value for x to check in the original
function. Try x  2.
f(x)  5x  f (2)  5 (2)  10
Substitute 10 into f1 (x). The output of the
inverse should be 2.
f 1( x ) 
x
10
 f 1(10) 
2
5
5
Think: (2, 10) in the original function should
be (10, 2) in the inverse. 
Use inverse operations to write the inverse of each function.
5. f (x)  x  4
________________________________________
7. f (x )  2x  3
________________________________________
6. f ( x ) 
x
6
________________________________________
8. f (x)  14x
________________________________________
4.3 Logarithmic Function
A logarithm is another way to work with exponents in equations.
If b x  a, then logb a  x.
Use the definition of the logarithm to write exponential equations in logarithmic form and to write logarithmic
equations in exponential form.
Exponential Form
Logarithmic Form
34  81
base, b  3
exponent, x  4
value, a  81
Logarithmic Form
log5 125  3
log3 81  4
Exponential Form
5 3  125
base, b  5
exponent, x  3
value, a  125
If no base is written for a logarithm, the base is assumed to be 10.
Example:
log 100  2 because 1 0 2  100.
Assume the base is 10.
Write each exponential equation in logarithmic form.
9. 7 2  49
10. 6 3  216
11. 2 5  32
Write each logarithmic equation in exponential form.
12. log9 729  3
13. log2 64  6
Complete the tables. Graph the functions.
15. f (x)  4 x
x
2
1
f (x)
1
16
1
4
f1(x)  log4 x
x
f 1(x)
1
16
1
4
0
1
2
14. log 1000  3
4.4 Properties of Logarithms
Product Property
The logarithm of a product can be written as the
sum of the logarithm of the numbers.
logb mn  logb m  logb n
where m, n, and b are all positive numbers and b  1
Simplify: log8 4  log8 16  log8 (4  16)  log8 64  2
]
Quotient Property
The logarithm of a quotient can be written as the logarithm of the
numerator minus the logarithm of the denominator.
m
logb  logb m  logb n
n
where m, n, and b are all positive numbers and b  1
 243 
log3 243  log3 9  log3 
  log3 27  3
9


Simplify:
Power Property
The logarithm of a power can be written as the product of the
exponent and the logarithm of the base.
logb a p  p logb a
for any real number p
where a and b are positive numbers and b  1
Simplify: log4 645  5 log4 64  5(3)  15
Inverse Properties
The logarithm of bx to the base b is equal
to x.
logb bx  x
b raised to the logarithm of x to the base b
is equal to x.
b logb x  x
The logarithm undoes the exponent when
the bases are the same.
Simplify: log7 74x  4x
The base of the log is 7 and the base of
the exponent is 7.
The exponent undoes the logarithm when
the bases are the same.
Simplify: 3log3 64  64
The base of the exponent is 3 and the
base of the log is 3.
Complete the steps to simplify each expression.
16. log6 54  log6 4
19. log6 65x
17. log2 128  log2 8
2. log2 164
18. log9 813
21. 2log2 3 x
4.5 Exponential and Logarithmic Equations and Inequalities
Solve 6x  2  500.
Step 1
Since the variable is in the exponent, take the log of both sides.
6x  2  500
log 6x  2  log 500
Use the Power Property of Logarithms: log ap  p log a.
Step 2
log 6x  2  log 500
(x  2) log 6  log 500
Step 3
“Bring down” the exponent to multiply.
Isolate the variable. Divide both sides by log 6.
(x  2) log 6  log 500
x2
Step 4
log 500
log 6
Solve for x. Subtract 2 from both sides.
x
Step 5
log 500
2
log 6
Use a calculator to approximate x.
x  1.468
Solve: log 80x  log 4  1
Step 1
Use the Quotient Property of Logarithms.
log 80x  log 4  1
log
Step 2
Simplify.
log
Step 3
80 x
1
4
80 x
1
4
log 20x  1
Use the definition of the logarithm:
if bx  a, then logb a  x.
log10 20x  1
Remember: Use 10 as the base
when the base is not given.
10 log10 20x = 101
20x = 101
Step 4
Solve for x. Divide both sides by 20.
10  20x
1
x
2
Solve and check.
22. 4x  32
23. 34x  90
24. 5x  3  600
25.log3 x4  8
26. log 4  log (x  2)  2
27. log 75x  log 3  1
4.6 The Natural Base, e
The natural logarithmic function, f (x)  ln x, is the inverse of the exponential function with the natural base
e, f (x)  ex.
The constant e is an irrational number. e  2.71828….
Properties of logarithms apply to the natural logarithm.
In particular:
ln 1  0
The base is e and e0  1.
ln e  1
Think: e1  e.
ln e x  x
e ln x  x
The natural logarithm and the
exponential function are inverses,
so they undo each other.
Use properties of logarithms to simplify expressions with e or “ln.”
Simplify: ln ex  2
Step 1 Use the Power Property.
“Bring down” the exponent
to multiply.
ln ex  2
(x  2) ln e
Simplify: e4 ln x
Step 1 Use the Power Property.
Write the exponent.
e 4 ln x
4
e ln x  x
eln x
Step 2 Simplify.
eln x
x4
ln e  1
Step 2 Simplify.
(x  2) ln e
x2
4
Simplify each expression.
28. ln e6x
29. ln e t  3
30. e 2 ln x
31.ln e1.8
32. ln ex  1
33. e7 ln x
4.7 Transforming Exponential and Logarithmic Functions
Translations
Vertical translation of f(x)
f(x)  k
Shift k units up for k  0.
Shift k units down for k  0.
f(x)  2x
g(x)  2x – 1
Think: k  1
y-intercept: y  20 1  1  1  0
Asymptote shifts down 1 unit: y  1
Horizontal translation of f (x)
f(x  h)
Shift h units right for h  0.
Shift h units left for h  0.
f(x)  2x
g(x)  2(x – 1)
Think: h  1
1
y-intercept: y  20  1  2 1  2
Asymptote does not shift: y  0
Translations
Vertical translation of f (x)
f(x)  k
Shift k units up for k  0.
Shift k units down for k  0.
f (x)  ln x
g(x)  ln x  1
Think: k  1
Asymptote does not shift: x  0.
ln 1  0, so (1, 1) lies on the graph.
Horizontal translation of f(x)
f (x  h)
Shift h units right for h  0.
Shift h units left for h  0.
f (x)  ln x
g (x)  ln (x  1) Think: h  1
Asymptote shifts left 1 unit: x  1.
When x  0, y  0. So, (0, 0) lies on
the graph.
Find the y-intercept and asymptote for each function. Then graph each function and
f (x)  2x on the same plane.
34. g(x)  2x  2
y-intercept: _______asymptote: _______
35. g(x)  2x  2
y-intercept: _______asymptote: _______
Reteach
Find the asymptote for each function. Then graph each function and
f (x)  ln x on the same plane.
36. g(x)  ln x  2
37. g(x)  ln (x  1)
Asymptote: ____________
Asymptote: ____________
4.8 Curve Fitting with Exponential and Logarithmic Models
Data Set 1
x
1
0
1
2
3
y
2.2
3
7
27
127
First
Difference
3  2.2
73
27  7
127  27
0.8
4
20
100
The first differences
seem to increase by
a constant factor.
Check the ratios of the first differences.
4
5
0.8
20
5
4
100
5
20
The data set is exponential, with a constant ratio of 5.
Data Set 2
x
1
0
1
2
3
y
4
1
2
5
8
1  (4)
First
Difference
3
2  (1)
52
85
3
3
3
The x-values increase by 1.
First differences are constant. The data set is linear, not exponential.
You can use the exponential regression feature on a graphing calculator to
model exponential data.
Use the exponential regression feature.
Select ExpReg [ENTER].
Then enter the lists: L1, L2
[ENTER].
Look for ExpReg in the STAT CALC menu.
38. a. Use exponential regression to find a function that models this data.
Time, t (min)
0
1
2
3
4
5
Bacteria, B(t)
250
360
485
686
964
1348
B(t)  ___________________
b. When will the number of bacteria reach 2000?
____________________