FLC
Ch 9
Math 335 Trigonometry
Sec 9.1: Exponential Functions
Properties of Exponents
∀𝑎 = 𝑏 > 0, 𝑏 ≠ 1 the following statements are true:

𝑏 𝑥 is a unique real number for all real numbers 𝑥

𝑓 (𝑥) = 𝑏 𝑥 is a function with domain (−∞, ∞)

𝑏𝑚 = 𝑏𝑛 iff 𝑚 = 𝑛 since 𝑓(𝑥) = 𝑏 𝑥 is a 1-1 function


If 𝑏 > 1 and 𝑚 < 𝑛, then 𝑏𝑚 < 𝑏𝑛 . That is, 𝑓 (𝑥) = 𝑏 𝑥 with 𝑏 > 1 is an increasing function.
If 0 < 𝑏 < 1 and 𝑚 < 𝑛, then 𝑏𝑚 > 𝑏𝑛 . That is, 𝑓 (𝑥) = 𝑏 𝑥 with 0 < 𝑏 < 1 is a decreasing
function.
Defn If 𝑏 > 0 𝑎𝑛𝑑 𝑏 ≠ 1 , then 𝑓(𝑥) = 𝑏 𝑥 is the exponential function with base 𝒃.
Characteristics of the Graph of 𝑓(𝑥) = 𝑏 𝑥
1
1) The points (−1, ) , (0,1), and (1, 𝑏) are on the graph.
𝑏
2) If 𝑏 > 1, then f is an increasing function; if 0 < 𝑏 < 1 , then 𝑓 if a decreasing function.
3) The 𝑥-axis is a horizontal asymptote.
4) The domain is (−∞, ∞) and the range is (0, ∞).
Ex 1 (# 4, #10)
Ex 2
1 𝑥
3
4
2
If 𝑓(𝑥) = 3𝑥 and 𝑔(𝑥) = ( ) , find 𝑓(−3) and 𝑔 ( ).
Graph each function. Label at least 3 points and include any pertinent information (e.g. asymptotes).
a) (# 14)
𝑓 (𝑥) = 4𝑥
b) (# 18)
2 𝑥
𝑓 (𝑥) = ( )
3
c) (# 24)
𝑓 (𝑥) = 2−|𝑥|
Page 1 of 10
FLC
Ch 9
1 𝑥
Ex 3
Sketch the graph of 𝑓 (𝑥) = (3) . Then refer to it and using graphing techniques to graph each
function.
a) (# 30)
b) (# 32)
1 𝑥
𝑓 (𝑥) = ( ) + 4
3
Ex 4
1 𝑥−4
𝑓(𝑥) = ( )
3
Solve each equation.
a) (# 48)
b) (# 54)
3𝑥−6
1
( )
2
52𝑝+1 = 25
c) (# 58)
3
( √5)
−𝑥
1
=( )
5
= 8𝑥+1
d)
𝑥+2
3𝑥
2 −4𝑥
=
1
27
Compound Interest
If 𝑃 dollars are deposited in an account paying an annual rate of interest of 𝑟 compounded (paid) 𝑛 times
per year, then after 𝑡 years the account will contain 𝐴 dollars, where
𝑟 𝑛𝑡
𝐴 = 𝑃 (1 + )
𝑛
𝑜𝑟
𝑟 𝑛𝑡
𝐹𝑉 = 𝑃𝑉 (1 + )
𝑛
Continuous Compounding
If 𝑃 dollars are deposited at a rate of interest 𝑟 compounded continuously for 𝑡 years, the compound
amount in dollars on deposit is 𝐴 = 𝑃𝑒 𝑟𝑡 .
Page 2 of 10
FLC
Ch 9
Ex 5 (# 64)
PP
Find the future value and interest earned if $56,780 is invested at 5.3% compounded
a) quarterly for 23 quarters
b) continuously for 15 years
Ex 6
Find the required annual interest rate to the nearest tenth of a percent to double your money if
interest is compounded quarterly for 8 years. What if it’s compounded continuously?
Sec 9.2: Logarithmic Functions
Ex
Is the inverse of 𝑓 (𝑥) = 𝑏 𝑥 a function? If so, find it.
Logarithm:
Meaning of 𝑦 = log 𝑏 𝑥
∀𝑦 ∈ ℝ 𝑎𝑛𝑑 ∀𝑏, 𝑥 ∈ ℝ+ 𝑤ℎ𝑒𝑟𝑒 𝑏 ≠ 1, 𝒚 = 𝐥𝐨𝐠 𝒃 𝒙 𝑖𝑓𝑓 𝒙 = 𝒃𝒚.
Logarithmic Function If 𝑏 > 0, 𝑏 ≠ 1, and𝑥 > 0, then 𝑓(𝑥) = log 𝑏 𝑥 defines the logarithmic function
with base 𝒃.
Page 3 of 10
FLC
Ch 9
Characteristics of the Graph of 𝑓(𝑥) = log 𝑏 𝑥
1
1) The points ( , −1) , (1, 0), and (𝑏, 1) are on the graph.
𝑏
2) If𝑏 > 1, then 𝑓 is an increasing function; if0 < 𝑏 < 1, then 𝑓 is a decreasing function.
3) The 𝑦-axis is a vertical asymptote.
4) The domain is (0, ∞) and the range is(−∞, ∞).
Properties of Logs
For 𝑥, 𝑦, 𝑏 > 0, 𝑏 ≠ 1, and any real number 𝑟:
𝑙𝑜𝑔𝑏 𝑥𝑦 = 𝑙𝑜𝑔𝑏 𝑥 + 𝑙𝑜𝑔𝑏 𝑦
Theorem on Inverses
𝑙𝑜𝑔𝑏
2 = 32
10
= 0.0001
b) (# 10)
log 5 5 = 1
a) (# 16)
b) (# 6)
−4
c)
1/5
𝑒
5
= √𝑒
Write an equivalent statement in exp form for each statement.
a) (# 8)
Ex 9
𝑏log𝑏 𝑥 = 𝑥 𝑓𝑜𝑟
𝑓𝑜𝑟
Write an equivalent statement in log form for each statement.
a) (# 4)
5
Ex 8
𝑙𝑜𝑔𝑏 𝑥 𝑟 = 𝑟 𝑙𝑜𝑔𝑏 𝑥
For 𝑏 > 0, 𝑏 ≠ 1,
log 𝑏 𝑏 𝑥 = 𝑥
Ex 7
𝑥
= 𝑙𝑜𝑔𝑏 𝑥 − 𝑙𝑜𝑔𝑏 𝑦
𝑦
1
log 4
= −3
64
c)
ln 𝑥 = 44
Solve each log equation.
1
𝑥 = log 6
216
b) (# 20)
𝑥 = 12
c) (# 28)
log12 5
4
𝑥 = log 5 √25
Ex 10 (# 34’)
Sketch the graph of 𝑓 (𝑥) = log 2 𝑥 then use graphing techniques to graph
𝑔(𝑥) = log 2 (𝑥 + 3). Find its domain and range. Next, find the domain and range of 𝑔(𝑥 − 10) + 2.
Page 4 of 10
FLC
Ch 9
Ex 11 (# 38’)
Sketch the graph of 𝑓 (𝑥) = log 1/2 𝑥 then use graphing techniques to graph
(
)
(
𝑔 𝑥 = |𝑙𝑜𝑔1/2 𝑥 − 2)|. Find its domain and range.
Ex 12 Use properties of logs to rewrite each expression. Simplify if possible and assume all variables
represent positive numbers.
a) (# 56)
log 2
b) (# 60)
3 𝑚 5 𝑛4
log 𝑝 √ 2
𝑡
2√3
5
Ex 13 Write each expression as a single logarithm with coefficient 1. Assume all variables represent
positive real numbers.
a)
(# 64)
1
2
log y p 3q 4  log y p 4 q3
2
3
Ex 14 (# 74)
b) (# 68) 
3
2
log 3 16 p 4  log 3 8 p3
4
3
Given log10 2  .3010 and log10 3  .4771 , find log10 361 3 .
Page 5 of 10
FLC
Ch 9
Sec 9.3: Evaluating Logarithms: Equations and Applications
The two most common bases for logarithms are 10 and e .
Common Logarithm
For all positive numbers x, log x  log10 x .
Natural Logarithm
For all positive numbers x, ln x  log e x .
Change-of-Base
For any positive real numbers 𝑥, 𝑎, and 𝑏, where a  1 and b  1 , log b x 
log a x
.
log a b
Property of Logarithms
If x, y, b  0 , b  1 , then x  y iff log b x  log b y .


Use a calculator to find an approximation to 4 decimal places of ln 2  e4 .
Ex 15 (# 22)
Find the pH of crackers with hydronium ion concentration of 3.9  109 . Use
pH   log H 3O , where H 3O  is the hydronium ion concentration (in moles per liter).
Ex 16 (# 24)
Ex 17 (# 40)




Use change-of-base and a calculator to find log
Ex 18 (# 46) Given f x   log 2 x , evaluate
 
a) f 23

b) f 2log2 2

19
5 to 4 decimal places.

c) f 22 log2 2

Page 6 of 10
FLC
Ch 9
I
, where I is
I0
the amplitude registered on a seismograph 100 km from the epicenter of the earthquake, and I 0 is the
Ex 19 (# 50)
The magnitude of an earthquake, measured on the Richter scale, is log10
amplitude of an earthquake of a certain (small) size. On June 16, 1999, the city of Puebla in central
Mexico was shaken by an earthquake that measured 6.7 on the Richter scale. Express this reading in
terms of I 0 .
Ex 20 Evaluate.
a) log 0.001 =
b) log 6 6 =
d) log 12 (−12) =
e) ln √𝑒 =
f) log 3 1 =
g) log 11 (117 ) =
h) log 5 5(−22) =
i) 3log3 (−4) =
Ex 21
c) log 4 256 =
13
Solve.
a)
2𝑥−7
8
c)
1
= 163𝑥−1
2𝑥−7
8 ∙ 10
b)
5𝑥+1 = 101−3𝑥
d)
=3
log(3𝑥 − 1) + log(3𝑥 + 2) = 2 log 2
Page 7 of 10
FLC
Ch 9
e)
f)
ln(5𝑥 − 2) + ln(7𝑥 − 1) = ln(5𝑥 + 37)
2 log 𝑥+7 (6) − log 𝑥+7 4 = 2
g)
h) [Hint: Multiply each side by 𝑒 𝑥 .]
4𝑥 + 2𝑥+2 − 12 = 0
𝑒 𝑥 − 𝑒 −𝑥
= −2
2
Practice Problems
[𝒔𝒊𝒏𝒉𝒙 = −𝟐]
y  2 x 4  1
1)
Graph the following exponential function.
2)
Solve the following exponential equations algebraically. Show all your work and give an exact answer.
a.
2
 
3
3)
Find the balance A for P dollars invested at rate r for t years and compounded n times.
a.
P = $2000, r = 6.5%, t = 10 years, compounded quarterly.
b.
P = $3000, r = 7%, t = 8 years, compounded continuously.
x

9
4
b.
2x  4  8x  6
3
c.
x 4  125
4)
Find the required annual interest rate to the nearest tenth of a percent for $48,000 to grow to $78,186.94
when compounded semiannually for 5 yr.
5)
a.
6)
Solve the following logarithmic equations algebraically. Show all your work and give an exact answer.
1
log x
 2
b. log x  5
c. log 4  x
3
16
16
Graph the following logarithmic function.
y = log 3 x  2  4
Page 8 of 10
FLC
Ch 9
7)
Use the properties of logarithms to expand or condense the following log expressions.
a.
log b
y3 x
z
1
2
2 log a x  log a y  log a z
3
3
b.
4
8)
Find the pH for lye soap which has a hydronium ion concentration of 3.210 14 moles per liter.
9)
Find the H 3 O  for drinking water which has a pH of 6.5.
10)
Use the change of base theorem to find an approximation for log 1 15 .


2
x
11)
Given f(x) = 3 , evaluate the following.
a.
f (log 3 7)
12)
a.
Find the decibel rating of a sound having an intensity of 100,000 I 0.
b.
If the intensity of the sound is doubled, by how much is the decibel rating increased?
a.
Find the Richter scale rating for an earthquake having an amplitude of 1,000,000 I 0.
b.
Express the magnitude of a 6.7 earthquake (on the Richter scale) in terms of I0.
13)
14)
15)
b.
f [log 3 (ln 3)]
c.
f [log 3 (2 ln 3)]
Solve.
a.
4 3x 1  6
a.
c.
2 x 5  3 x
 .06 
d. 10001 

4 

c.
6e 4x 3  1  11
log (2x – 1) + log 10x = log 10
b.
ln x + ln (x – 1) = 2
log 6 4x  log 6 x  3  log 6 12
d.
log (11x + 9) = 3 + log (x + 3)
b.
4t
 5000
16) Elena McDuff wants to buy an $8,000 painting. She has saved $6,000. Find the number of years (to the
nearest tenth) it will take for her savings to grow to $8,000 at 5.7% compounded monthly.
17) At what interest rate will $2,500 grow to $4,671.04 if invested for 10 years and interest is compounded
quarterly?
Answers
1.
HA: y = 1
x
2
3
4
5
6
“key point”: (4, 0)
y
3
4
1
2
0
-1
-3
2.
a.
x=2
b.
x = 11
3.
a.
$3811.12
b.
$5252.02
4.
10%
c.
x = 625
Page 9 of 10
FLC
Ch 9
5.
a.
6.
VA: x = –2; exponential form of the equation: x  3 y  4  2
7.
a.
x=4
b.
x
 17
9
y
-6
 53
-5
-1
1
7
-4
-3
-2
3 log b y 
1
log b x  4 log b z
2
x = 243
c.
b.
log a
x
1
2
x2
3
8.
13.5
9.
3.2 10 7
10.
–3.9069
11.
a.
7
b.
ln 3  1.0986
12.
a.
50
b.
about 3 decibels
13.
a.
6
b.
about 5,000,000 I0
14.
a.
.7642
b.
15.
a.
1
b.
16.
5.1 years
17.
6.3%
yz 2
c.
2 ln 3 = ln 32 = ln 9  2.1972
8.5476
c.
–.5767
d.
27.0246
3.2639
c.
4.5
d.
Ø
Page 10 of 10