Chapter 9 Roots, Powers and Logarithms
9.1 𝒏th Roots
Ex 1: Evaluate the following without the use of a calculator
a. 32
b. 33
c. 34
Ex 2: Evaluate the following without the use of a calculator
a. √9
3
b. √27
c.
4
√81
𝑛th roots
Ex 3: Evaluate the following without the use of a calculator
3
a. √125𝑥 3
4
b. √64𝑥 5 𝑦 7
3
256
c. √27𝑥 3
Expressing Roots as Powers
Ex 3: Evaluate the following without the use of a calculator
1
a. 643
1 1
b. (36)2
1
c. −164
1
d. (−243)3
Exponential Growth or Decay
Ex 4: A bowling game costs 50 cents in 1960 might cost $3.00 in 2010. What is the annual growth?
9.2 Rational Exponents
Rational Powers
Ex 1: Evaluate the following without the use of a calculator
2
a. 643
3
b. 1212
4 5
c. (9)2
1
d. (125𝑥 3 𝑦 4 )2
Properties of Powers
Property
Power of a Power
Product of Powers
Quotient of Powers
Power of a Product
Power of a Quotient
Zero Exponent
Negative Exponent
Variables
Example
Ex 2: Use the properties of powers to evaluate the following
a. 𝑥 6 ∙ 𝑥 5
b. 4𝑥 7 ∙ 6𝑥 4
72𝑥 7
c. ( 6𝑥 2 )
6𝑥 5
d. ( 3𝑥 )3
e. (
20𝑥 7 −2
)
6𝑥
Ex 3: Solve for x
1
a. 3𝑥 = 9
5
27
b. (3)𝑥 = 125
1
c. (3)𝑥 = 243
d. 5 ∙ 3𝑥 = 5
e. 16𝑥 = 2
9.3 Logarithm Functions
Definition of Logarithm
Ex 1: Rewrite in logarithmic/exponential form
a. 𝑙𝑜𝑔2 16 = 4
b. 𝑙𝑜𝑔1 8 = −3
2
c. 32 = 9
d.
14
3
1
= 81
Ex 2: Evaluate the following
a. log base 8 of 512
1
b. 𝑙𝑜𝑔8 64
7
c. 𝑙𝑜𝑔8 √8
d. 𝑙𝑜𝑔4 2
Common Logarithms
Ex 3: Evaluate the following common logarithms
a. log 100
b. log 0.00001
c. log 10√10
Common Logs to Solve Equations
Ex 4: Solve 10𝑥 = 7
Ex 5: Solve log 𝑥 = 2.873
9.4 e and Natural Logarithms
Recall the Compound Interest Formula
Let P = $1 for 1 year and 100% interest rate.
Compound Schedule
𝒏
Every day
365
Every hour
8760
Every minute
525,600
Every second
31,536,000
𝒓
𝑨 = 𝑷(𝟏 + )𝒏𝒕
𝒏
Balance in $ after 1 year
The number e
Compounding Continuously/ Annual Yield
Ex 1: If $500 is put in a saving account at an 8% annual rate compounded continuously, calculate
a. The annual yield
b. The value of the investment after one year
Ex 2: Suppose $500 is invested in an account paying an 8% annual interest rate.
a. Give the balance after 4.5 years if interest is compounded continuously.
b. How does the balance compare with quarterly compounding?
Natural Logarithms
Ex 3: Convert to logarithmic/exponential form
a. 𝑙𝑛2 = 𝑥
b. 𝑒 2𝑥 = 10
Ex 4: Evaluate the following
a. ln 𝑒 4
b. ln 𝑒
c. log(ln 𝑒 10 )
Ex 5: Under certain conditions, the height h in feet above sea level can be approximated from the
ln 𝑃−ln 14.7
atmospheric pressure P in pounds per square inch (psi) using the equation ℎ = −0.000039 . Human blood
“boils” at 0.9 psi, and such a situation is fatal. At what height above sea level will blood “boil” in an
unpressurized airplane cabin?
9.5 Properties of Logarithms
Properties of Logs
Property
Variables
Example
Logarithm of 1
Logarithm of the Base
Logarithm of a Power
Logarithm of a Product
Logarithm of a Qutient
Ex 1: Using log 2 ≈ 0.301 and log 3 ≈ 0.477, compute the following without a calculator
a. log 6
b. log 20
c. log 50
Ex 3: Evaluate the following using properties of logs without a calculator
3
a. 𝑙𝑜𝑔5 √5
b. ln 𝑒 6 − ln 𝑒 2
c. 𝑙𝑜𝑔2 (−16)
d. 𝑙𝑜𝑔5 75 − 𝑙𝑜𝑔5 3
Ex 4: Rewrite as a single logarithm. If you can evaluate, find a value for the logarithm.
a. log 40 + 2 log 5
b. ln 14 − ln 2 − ln 7
c. 3𝑙𝑜𝑔3 𝑥 + 4𝑙𝑜𝑔3 𝑦 − 4𝑙𝑜𝑔3 𝑧
Ex 5: Rewrite as multiple logarithms. If you can evaluate, find the value of the logarithm.
3
a. 𝑙𝑜𝑔3 81
b. ln 𝑥𝑦𝑧 2
1
c. 𝑙𝑛 𝑒
𝑥2
d. 𝑙𝑜𝑔√𝑦 3
9.6 Solving Exponential Equations
Using logs to solve Exponential Equations
Ex 1: Solve the following
a. 5𝑥 = 46
b. 23−𝑥 = 565
c. 8(10𝑥 ) = 12
d. 𝑒 3𝑥 = 12
e. 7 − 2𝑒 𝑥 = 5
Ex 2: A family has $34,500 in a savings account that is paying interest at a rate of 0.71%. If the interest is
compounding continuously, how long would it take to grow to $40,000?
Half Life
Ex 3: Carbon-14 has a half-life of 5730 years. From this assumption, find the approximate age of a skull found
by an archeologist if the skull has 67% of its original carbon-14 concentration.
Change of base formula
Ex 4: Use the change of base formula to evaluate the following logs
a. 𝑙𝑜𝑔5 71
b. 𝑙𝑜𝑔12 600