Exponential and Logarithmic Functions Solving Logarithm Properties InversesApplicationGraphing 10 20...
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Exponential and Logarithmic Functions Solving Logarithm Properties Inverses Application Graphing 10 10 10 10 10 20 20 20 20 20 30 30 30 30 30 40 40 40 40 40 50 50 50 50 50
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Transcript of Exponential and Logarithmic Functions Solving Logarithm Properties InversesApplicationGraphing 10 20...
Exponential and Logarithmic FunctionsSolving
Logarithm Properties Inverses Application Graphing
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Solve, round to nearest hundredth
Answer
52𝑥+8=125𝑥
52𝑥+8=125𝑥
52𝑥+8=53 𝑥
2 𝑥+8=3𝑥8=𝑥
Solve, round to nearest hundredth
Answer
7 (5𝑥)=168
7 (5𝑥)=168
5𝑥=24𝑥=log 524
𝑥=log 24log 5
≈1.97
Solve, round to nearest hundredth
Answer
63𝑥−20=3
63𝑥=233 𝑥=log6 23
3 𝑥=log 23log 6
3 𝑥≈1.75𝑥≈0.58
Solve, round to nearest hundredth
Answer
3+ log 4(𝑥−7 )=5
3+ log 4(𝑥−7 )=5
log 4(𝑥−7)=2
𝑥−7=42
𝑥−7=16
𝑥=23
Solve, round to nearest hundredth
Answer
log (𝑥+3)− log 4=3
log (𝑥+3)− log 4=3
log𝑥+3
4=3
𝑥+34
=103
𝑥+34
=1000
𝑥+3=4000
𝑥=3997
Write in logarithm form
Answer
𝑦=7𝑥
log 7 𝑦=𝑥
Write in exponential form
Answer
𝑦=log3𝑥
3𝑦=𝑥
Evaluate each of the expressions
Answer
log 18
log 517
log 4 64
log 18
log 517
log 4 64
≈1.256
≈1.760
¿3
Simplify to a single logarithm
Answer
2 log𝑎−3 log𝑏+4 log𝑐
2 log𝑎−3 log𝑏+4 log𝑐
log 𝑎2−log𝑏3+log𝑐4
log𝑎2
𝑏3 + log𝑐4
log𝑎2𝑐4
𝑏3
Expand the expression
Answer
log2𝑎3
𝑏4
log2𝑎3
𝑏4
log 2𝑎3−log𝑏4
log 2+ log𝑎3− log𝑏4
log 2+3 log𝑎−4 log𝑏
Find the inverse.
Answer
𝑦=¿
𝑦=¿
𝑥=¿
𝑥+4=¿log 5(𝑥+4)=𝑦+3
log 5(𝑥+4)−3=𝑦
Find the inverse.
Answer
𝑦=7 (2)𝑥+5
𝑦=7 (2)𝑥+5
𝑥=7 (2)𝑦+5
𝑥7=(2)𝑦+5
log 2𝑥7=𝑦+5
log 2𝑥7−5=𝑦
Find the inverse.
Answer
𝑦=log8𝑥−7
𝑦=log8𝑥−7
𝑥=log 8 𝑦−7
𝑥+7=log8 𝑦
8𝑥+7=𝑦
Find the inverse.
Answer
𝑦=4 log(3 𝑥+7)
𝑦=4 log(3 𝑥+7)𝑥=4 log (3 𝑦+7)
𝑥4=log(3 𝑦+7)
10𝑥4 =3 𝑦+7
10𝑥4−7=3 𝑦
10𝑥4−73
=𝑦
Find the inverse.
Answer
𝑦=13
ln(𝑥+5)−2
𝑦=13
ln(𝑥+5)−2
𝑥=13
ln(𝑦+5)−2
𝑥+2=13
ln(𝑦+5)
3 (𝑥+2)=ln(𝑦+5)
𝑒3 (𝑥+2)=𝑦 +5
𝑒3 (𝑥+2)−5=𝑦
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 No money is added or withdrawn form the account.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.
2. How much will the account be worth in 5 years?
3. How many years until the account doubles?
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 No money is added or withdrawn form the account.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?
𝑦=1500 (1+.06)𝑥
𝑦=1500 (1+.06)5¿2007.34
3000=1500 (1+.06)𝑥12 years
𝑥=log 1.06 2=11.896
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.
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=1570 (1.2)𝑥
𝑏=18841570
=1.2 𝑦=1570 (1.2)9
8,100 bears
𝑦=𝑎(𝑏)𝑥
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.
2. How many years until the population goes below 4 million?
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?
𝑃=7.3(1−0.023)𝑡
4=7.3(1−0.023)𝑡
26 years𝑡=log 0.977(0.5479)=25.854
Answer
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 = ae0.00012t, 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?
𝑦=𝑎𝑒− 0.00012𝑡
32=100𝑒− 0.00012𝑡
0.32=𝑒−0.00012𝑡
ln 0.32=−0.00012𝑡ln 0.32−0.00012
=𝑡
years
Answer
A new truck that sells for $29,000 depreciates 12% each year. What is the value of the truck after 7 years?
𝑦=29000 (1−0.12)𝑥
𝑦=29000 (1−0.12)7
𝑦=11,851.59
$11,851.59
-5 5-5
5
x
yGraph and Identify the domain and range
Answer
𝑦=2𝑥−2−3
-5 5-5
5
x
y
Domain: All real numbers
Range:
𝑦=2𝑥−2−3
-5 5-5
5
x
yGraph and Identify the domain and range
Answer
𝑦=2 (2 )𝑥− 3+1
-5 5-5
5
x
y
Domain: All real numbers
Range:
𝑦=2 (2 )𝑥− 3+1
-5 5-5
5
x
yGraph and Identify the domain and range
Answer
𝑦=log3(𝑥+1)+2
-5 5-5
5
x
y
Domain:
Range: All real numbers
𝑦=log3(𝑥+1)+2
-5 5-5
5
x
yGraph and Identify the domain and range
Answer
𝑦=2 log 5(𝑥)−3
-5 5-5
5
x
y
Domain:
Range: All real numbers
𝑦=2 log 5(𝑥)−3
-5 5-5
5
x
yGraph and Identify the domain and range
Answer
𝑦=−3 (2 )𝑥+1+2
-5 5-5
5
x
y
𝑦=−3 (2 )𝑥+1+2
Domain: All real numbers
Range:
