Exponential and Logarithmic Functions Solving Logarithm Properties Inverses Application Graphing 10 20 30 40 50

Solve, round to nearest hundredth 5 2𝑥+8 = 125 𝑥 Answer

5 2𝑥+8 = 125 𝑥 5 2𝑥+8 = 5 3𝑥 2𝑥+8=3𝑥 8=𝑥

Solve, round to nearest hundredth 7( 5 𝑥 )=168 Answer

7( 5 𝑥 )=168 5 𝑥 =24 𝑥= log 5 24 𝑥= log 24 log 5 ≈1.97

Solve, round to nearest hundredth 6 3𝑥 −20=3 Answer

6 3𝑥 =23 3𝑥= log 6 23 3𝑥= log 23 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= 4 2 𝑥−7=16 𝑥=23

Solve, round to nearest hundredth log (𝑥+3) − log 4 =3 Answer

log (𝑥+3) − log 4 =3 log 𝑥+3 4 =3 𝑥+3=4000 𝑥=3997 𝑥+3 4 = 10 3 𝑥+3 4 =1000

Write in logarithm form 𝑦= 7 𝑥 Answer

log 7 𝑦 =𝑥

Write in exponential form 𝑦= log 3 𝑥 Answer

3 𝑦 =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 𝑎 2 − log 𝑏 3 + log 𝑐 4 log 𝑎 2 𝑏 3 + log 𝑐 4 log 𝑎 2 𝑐 4 𝑏 3

Expand the expression log 2 𝑎 3 𝑏 4 Answer

log 2 𝑎 3 𝑏 4 log 2 𝑎 3 − log 𝑏 4 log 2 + log 𝑎 3 − 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 log 2 𝑥 7 −5=𝑦 𝑥 7 = (2) 𝑦+5 log 2 𝑥 7 =𝑦+5

Find the inverse. 𝑦= log 8 𝑥−7 Answer

𝑦= log 8 𝑥−7 𝑥= log 8 𝑦−7 𝑥+7= log 8 𝑦 8 𝑥+7 =𝑦

Find the inverse. 𝑦=4 log (3𝑥+7) Answer

𝑦=4 log (3𝑥+7) 𝑥=4 log (3𝑦+7) 10 𝑥 4 −7 3 =𝑦 𝑥 4 = log (3𝑦+7) 10 𝑥 4 =3𝑦+7 10 𝑥 4 −7=3𝑦

Find the inverse. 𝑦= 1 3 ln (𝑥+5) −2 Answer

𝑦= 1 3 ln (𝑥+5) −2 𝑒 3(𝑥+2) =𝑦+5 𝑥= 1 3 ln (𝑦+5) −2 𝑒 3(𝑥+2) −5=𝑦 𝑥+2= 1 3 ln (𝑦+5) 3(𝑥+2)= ln (𝑦+5)

Suppose you deposit $1500 in a savings account that pays 6% Suppose you deposit $1500 in a savings account that pays 6%. No money is added or withdrawn form the account. Write an equation to model this situation. How much will the account be worth in 5 years? How many years until the account doubles? Answer

Suppose you deposit $1500 in a savings account that pays 6% Suppose you deposit $1500 in a savings account that pays 6%. No money is added or withdrawn form the account. Write an equation to model this situation. How much will the account be worth in 5 years? How many years until the account doubles? 𝑦=1500 (1+.06) 𝑥 𝑦=1500 (1+.06) 5 =2007.34 3000=1500 (1+.06) 𝑥 12 years 𝑥= log 1.06 2 =11.896

In 2009, there were 1570 bears in a wildlife refuge 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 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 𝑏= 1884 1570 =1.2 𝑦=1570 (1.2) 9 8,100 bears

Suppose the population of a country is currently 7. 3 million people 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. Write an equation to model this situation. How many years until the population goes below 4 million? Answer

Suppose the population of a country is currently 7. 3 million people 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. Write an equation to model this situation. How many years until the population goes below 4 million? 𝑃=7.3 (1−0.023) 𝑡 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=𝑎 𝑒 −0.00012𝑡 32=100 𝑒 −0.00012𝑡 0.32= 𝑒 −0.00012𝑡 ln 0.32 =−0.00012𝑡 ln 0.32 −0.00012 =𝑡 𝑡=9,496 years

A new truck that sells for $29,000 depreciates 12% each year 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= 2 𝑥−2 −3 Answer

𝑦= 2 𝑥−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