Transcript Unit Review
Exponential and Logarithmic Functions
Solving
Logarithm
Properties
Inverses
Application
Graphing
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Solve, round to nearest hundredth
2๐ฅ+8
5
= 125
Answer
๐ฅ
52๐ฅ+8 = 125๐ฅ
52๐ฅ+8 = 53๐ฅ
2๐ฅ + 8 = 3๐ฅ
8=๐ฅ
Solve, round to nearest hundredth
7(5๐ฅ ) = 168
Answer
7(5๐ฅ ) = 168
5๐ฅ = 24
๐ฅ = log 5 24
log 24
๐ฅ=
โ 1.97
log 5
Solve, round to nearest hundredth
63๐ฅ โ 20 = 3
Answer
63๐ฅ = 23
3๐ฅ = log 6 23
log 23
3๐ฅ =
log 6
3๐ฅ โ 1.75
๐ฅ โ 0.58
Solve, round to nearest hundredth
3 + log 4 (๐ฅ โ 7) = 5
Answer
3 + log 4 (๐ฅ โ 7) = 5
log 4 (๐ฅ โ 7) = 2
๐ฅ โ 7 = 42
๐ฅ โ 7 = 16
๐ฅ = 23
Solve, round to nearest hundredth
log(๐ฅ + 3) โ log 4 = 3
Answer
log(๐ฅ + 3) โ log 4 = 3
๐ฅ+3
log
=3
4
๐ฅ+3
= 103
4
๐ฅ+3
= 1000
4
๐ฅ + 3 = 4000
๐ฅ = 3997
Write in logarithm form
๐ฆ = 7๐ฅ
Answer
log 7 ๐ฆ = ๐ฅ
Write in exponential form
๐ฆ = log 3 ๐ฅ
Answer
3๐ฆ = ๐ฅ
Evaluate each of the expressions
log 18
log 5 17
log 4 64
Answer
log 18 โ 1.256
log 5 17 โ 1.760
log 4 64 = 3
Simplify to a single logarithm
2 log ๐ โ 3 log ๐ + 4 log ๐
Answer
2 log ๐ โ 3 log ๐ + 4 log ๐
log ๐2 โ log ๐ 3 + log ๐ 4
๐2
log 3 + log ๐ 4
๐
๐2 ๐ 4
log 3
๐
Expand the expression
2๐3
log 4
๐
Answer
2๐3
log 4
๐
3
log 2๐ โ log ๐
3
4
log 2 + log ๐ โ log ๐
4
log 2 + 3 log ๐ โ 4 log ๐
Find the inverse.
๐ฆ = (5)๐ฅ+3 โ 4
Answer
๐ฆ = (5)๐ฅ+3 โ 4
๐ฅ = (5)๐ฆ+3 โ 4
๐ฅ + 4 = (5)๐ฆ+3
log 5 (๐ฅ + 4) = ๐ฆ + 3
log 5 (๐ฅ + 4) โ 3 = ๐ฆ
Find the inverse.
๐ฆ = 7(2)๐ฅ+5
Answer
๐ฆ = 7(2)๐ฅ+5
๐ฅ = 7(2)๐ฆ+5
๐ฅ
= (2)๐ฆ+5
7
๐ฅ
log 2 = ๐ฆ + 5
7
๐ฅ
log 2 โ 5 = ๐ฆ
7
Find the inverse.
๐ฆ = log 8 ๐ฅ โ 7
Answer
๐ฆ = log 8 ๐ฅ โ 7
๐ฅ = log 8 ๐ฆ โ 7
๐ฅ + 7 = log 8 ๐ฆ
8๐ฅ+7 = ๐ฆ
Find the inverse.
๐ฆ = 4 log(3๐ฅ + 7)
Answer
๐ฆ = 4 log(3๐ฅ + 7)
๐ฅ = 4 log(3๐ฆ + 7)
๐ฅ
= log(3๐ฆ + 7)
4
๐ฅ
104
= 3๐ฆ + 7
๐ฅ
104
โ 7 = 3๐ฆ
๐ฅ
104
โ7
=๐ฆ
3
Find the inverse.
1
๐ฆ = ln(๐ฅ + 5) โ 2
3
Answer
1
๐ฆ = ln(๐ฅ + 5) โ 2
3
1
๐ฅ = ln(๐ฆ + 5) โ 2
3
1
๐ฅ + 2 = ln(๐ฆ + 5)
3
3(๐ฅ + 2) = ln(๐ฆ + 5)
๐ 3(๐ฅ+2) = ๐ฆ + 5
๐ 3(๐ฅ+2) โ 5 = ๐ฆ
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money is added or withdrawn form the account.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money is added or withdrawn form the account.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.
๐ฆ = 1500(1 + .06)๐ฅ
2. How much will the account be worth in 5 years?
๐ฆ = 1500(1 + .06)5 = 2007.34
3. How many years until the account doubles?
3000 = 1500(1 + .06)๐ฅ
๐ฅ = 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.
๐ฆ = ๐(๐)๐ฅ
๐ฆ = 1570(1.2)๐ฅ
1884
๐=
= 1.2
1570
๐ฆ = 1570(1.2)9
8,100 bears
Suppose the population of a country is currently 7.3 million people.
Studies show this countryโs 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 countryโs population is declining at a rate of 2.3%
each year.
1. Write an equation to model this situation.
๐ = 7.3(1 โ 0.023)๐ก
2. How many years until the population goes below 4 million?
4 = 7.3(1 โ 0.023)๐ก
๐ก = 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๏ญ0.00012t, 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
๐ฆ = ๐๐ โ0.00012๐ก
32 = 100๐ โ0.00012๐ก
0.32 = ๐ โ0.00012๐ก
ln 0.32 = โ0.00012๐ก
ln 0.32
=๐ก
โ0.00012
๐ก = 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
๐ฆ = 29000(1 โ 0.12)๐ฅ
๐ฆ = 29000(1 โ 0.12)7
๐ฆ = 11,851.59
$11,851.59
Graph and Identify the domain and range
๐ฆ = 2๐ฅโ2 โ 3
Answer
๐ฆ = 2๐ฅโ2 โ 3
Domain: All real numbers
Range: ๐ฆ > โ3
Graph and Identify the domain and range
๐ฆ=2 2
๐ฅโ3
+1
Answer
๐ฆ=2 2
๐ฅโ3
+1
Domain: All real numbers
Range: ๐ฆ > 1
Graph and Identify the domain and range
๐ฆ = log 3 (๐ฅ + 1) + 2
Answer
๐ฆ = log 3 (๐ฅ + 1) + 2
Domain: ๐ฅ > โ1
Range: All real numbers
Graph and Identify the domain and range
๐ฆ = 2 log 5 (๐ฅ) โ 3
Answer
๐ฆ = 2 log 5 (๐ฅ) โ 3
Domain: ๐ฅ > 0
Range: All real numbers
Graph and Identify the domain and range
๐ฆ = โ3 2
๐ฅ+1
+2
Answer
๐ฆ = โ3 2
๐ฅ+1
+2
Domain: All real numbers
Range: ๐ฆ < 2