Exponential and Logarithmic Functions
Solving
Logarithm
Properties
Inverses
Application
Graphing
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Solve, round to nearest hundredth
2𝑥+8
5
= 125
Answer
𝑥
52𝑥+8 = 125𝑥
52𝑥+8 = 53𝑥
2𝑥 + 8 = 3𝑥
8=𝑥
Solve, round to nearest hundredth
7(5𝑥 ) = 168
Answer
7(5𝑥 ) = 168
5𝑥 = 24
𝑥 = log 5 24
log 24
𝑥=
≈ 1.97
log 5
Solve, round to nearest hundredth
63𝑥 − 20 = 3
Answer
63𝑥 = 23
3𝑥 = log 6 23
log 23
3𝑥 =
log 6
3𝑥 ≈ 1.75
𝑥 ≈ 0.58
Solve, round to nearest hundredth
3 + log 4 (𝑥 − 7) = 5
Answer
3 + log 4 (𝑥 − 7) = 5
log 4 (𝑥 − 7) = 2
𝑥 − 7 = 42
𝑥 − 7 = 16
𝑥 = 23
Solve, round to nearest hundredth
log(𝑥 + 3) − log 4 = 3
Answer
log(𝑥 + 3) − log 4 = 3
𝑥+3
log
=3
4
𝑥+3
= 103
4
𝑥+3
= 1000
4
𝑥 + 3 = 4000
𝑥 = 3997
Write in logarithm form
𝑦 = 7𝑥
Answer
log 7 𝑦 = 𝑥
Write in exponential form
𝑦 = log 3 𝑥
Answer
3𝑦 = 𝑥
Evaluate each of the expressions
log 18
log 5 17
log 4 64
Answer
log 18 ≈ 1.256
log 5 17 ≈ 1.760
log 4 64 = 3
Simplify to a single logarithm
2 log 𝑎 − 3 log 𝑏 + 4 log 𝑐
Answer
2 log 𝑎 − 3 log 𝑏 + 4 log 𝑐
log 𝑎2 − log 𝑏 3 + log 𝑐 4
𝑎2
log 3 + log 𝑐 4
𝑏
𝑎2 𝑐 4
log 3
𝑏
Expand the expression
2𝑎3
log 4
𝑏
Answer
2𝑎3
log 4
𝑏
3
log 2𝑎 − log 𝑏
3
4
log 2 + log 𝑎 − log 𝑏
4
log 2 + 3 log 𝑎 − 4 log 𝑏
Find the inverse.
𝑦 = (5)𝑥+3 − 4
Answer
𝑦 = (5)𝑥+3 − 4
𝑥 = (5)𝑦+3 − 4
𝑥 + 4 = (5)𝑦+3
log 5 (𝑥 + 4) = 𝑦 + 3
log 5 (𝑥 + 4) − 3 = 𝑦
Find the inverse.
𝑦 = 7(2)𝑥+5
Answer
𝑦 = 7(2)𝑥+5
𝑥 = 7(2)𝑦+5
𝑥
= (2)𝑦+5
7
𝑥
log 2 = 𝑦 + 5
7
𝑥
log 2 − 5 = 𝑦
7
Find the inverse.
𝑦 = log 8 𝑥 − 7
Answer
𝑦 = log 8 𝑥 − 7
𝑥 = log 8 𝑦 − 7
𝑥 + 7 = log 8 𝑦
8𝑥+7 = 𝑦
Find the inverse.
𝑦 = 4 log(3𝑥 + 7)
Answer
𝑦 = 4 log(3𝑥 + 7)
𝑥 = 4 log(3𝑦 + 7)
𝑥
= log(3𝑦 + 7)
4
𝑥
104
= 3𝑦 + 7
𝑥
104
− 7 = 3𝑦
𝑥
104
−7
=𝑦
3
Find the inverse.
1
𝑦 = ln(𝑥 + 5) − 2
3
Answer
1
𝑦 = ln(𝑥 + 5) − 2
3
1
𝑥 = ln(𝑦 + 5) − 2
3
1
𝑥 + 2 = ln(𝑦 + 5)
3
3(𝑥 + 2) = ln(𝑦 + 5)
𝑒 3(𝑥+2) = 𝑦 + 5
𝑒 3(𝑥+2) − 5 = 𝑦
Suppose you deposit $1500 in a savings account that pays 6%. No
money is added or withdrawn form the account.
1. Write an equation to model this situation.
2. How much will the account be worth in 5 years?
3. How many years until the account doubles?
Answer
Suppose you deposit $1500 in a savings account that pays 6%. No
money is added or withdrawn form the account.
1. Write an equation to model this situation.
𝑦 = 1500(1 + .06)𝑥
2. How much will the account be worth in 5 years?
𝑦 = 1500(1 + .06)5 = 2007.34
3. How many years until the account doubles?
3000 = 1500(1 + .06)𝑥
𝑥 = log1.06 2 = 11.896
12 years
In 2009, there were 1570 bears in a wildlife refuge. In 2010
approximately 1884 bears. If this trend continues and the bear
population is increasing exponentially, how many bears will there
be in 2018?
Write an exponential function to model the situation, then solve.
Answer
In 2009, there were 1570 bears in a wildlife refuge. In 2010
approximately 1884 bears. If this trend continues and the bear
population is increasing exponentially, how many bears will there
be in 2018?
Write an exponential function to model the situation, then solve.
𝑦 = 𝑎(𝑏)𝑥
𝑦 = 1570(1.2)𝑥
1884
𝑏=
= 1.2
1570
𝑦 = 1570(1.2)9
8,100 bears
Suppose the population of a country is currently 7.3 million people.
Studies show this country’s population is declining at a rate of 2.3%
each year.
1. Write an equation to model this situation.
2. How many years until the population goes below 4 million?
Answer
Suppose the population of a country is currently 7.3 million people.
Studies show this country’s population is declining at a rate of 2.3%
each year.
1. Write an equation to model this situation.
𝑃 = 7.3(1 − 0.023)𝑡
2. How many years until the population goes below 4 million?
4 = 7.3(1 − 0.023)𝑡
𝑡 = log 0.977 (0.5479) = 25.854
26 years
By measuring the amount of carbon-14 in an object, a
paleontologist can determine its approximate age. The amount of
carbon-14 in an object is given by y = ae0.00012t, where a is the
amount of carbon-14 originally in the object, and t is the age of
the object in years.
A fossil of a bone contains 32% of its original carbon-14. What is the
approximate age of the bone?
Answer
𝑦 = 𝑎𝑒 −0.00012𝑡
32 = 100𝑒 −0.00012𝑡
0.32 = 𝑒 −0.00012𝑡
ln 0.32 = −0.00012𝑡
ln 0.32
=𝑡
−0.00012
𝑡 = 9,496 years
A new truck that sells for $29,000 depreciates 12% each
year. What is the value of the truck after 7 years?
Answer
𝑦 = 29000(1 − 0.12)𝑥
𝑦 = 29000(1 − 0.12)7
𝑦 = 11,851.59
$11,851.59
Graph and Identify the domain and range
𝑦 = 2𝑥−2 − 3
Answer
𝑦 = 2𝑥−2 − 3
Domain: All real numbers
Range: 𝑦 > −3
Graph and Identify the domain and range
𝑦=2 2
𝑥−3
+1
Answer
𝑦=2 2
𝑥−3
+1
Domain: All real numbers
Range: 𝑦 > 1
Graph and Identify the domain and range
𝑦 = log 3 (𝑥 + 1) + 2
Answer
𝑦 = log 3 (𝑥 + 1) + 2
Domain: 𝑥 > −1
Range: All real numbers
Graph and Identify the domain and range
𝑦 = 2 log 5 (𝑥) − 3
Answer
𝑦 = 2 log 5 (𝑥) − 3
Domain: 𝑥 > 0
Range: All real numbers
Graph and Identify the domain and range
𝑦 = −3 2
𝑥+1
+2
Answer
𝑦 = −3 2
𝑥+1
+2
Domain: All real numbers
Range: 𝑦 < 2