Chapter 11 Exponential and Logarithmic Functions
11.1 Real Exponents
Properties
Property
Product
Power of a Power
Power of a Quotient
Power of a Product
Quotient
Rational Exponents
Negative Exponents
Ex 1: Simplify/Evaluate each (no calculator)
4
a. 83
b. 16𝟐𝟕 ∙ 8𝟐
c.
d.
493 ∙3432
710
(36 )(913 )
(√27)9
Definition
Example
Rewriting between Exponential and Radical Form
Ex 2: Write the following using rational exponents
10
a. √𝑓 15 𝑟 20 𝑖 78
3
b. √64𝑠 9 𝑡15
Ex 3: Write the following using a radical
5
1
2
2
1
a. 𝑑 7 𝑎4 𝑦 3
b. 12𝑥 3 𝑦 2
3
Ex 4: Solve 734 = 𝑥 4 + 5
11.2 Exponential Functions
Parent Function:
Horizontal Shift:
Vertical Shift:
Horizontal Stretch/Compression:
Vertical Stretch/Compression:
Ex 1: Graph the following
a. 𝑦 = 2𝑥
1
b. 𝑓(𝑥) = 2(2)𝑥
c. 𝑦 = 3(2𝑥 )
1
d. 𝑦 = −2( )𝑥+1
2
Exponential Growth/Decay Model
Ex 2: Mrs. A bought a car in 2006 for $15,000. The value of cars depreciate exponentially at a rate of 15% a
year. What is the value of her car in 2015?
11.4 Logarithmic Functions and Common Logarithm
Exponential Form
Logarithmic Form
Ex 1: Rewrite the following in exponential/logarithmic form
a.
2𝑥
3
= 10
b. log 4 𝑥 = 2
Evaluating Logs Without a Calculator
Ex 1: Evaluate the following
1
a. log 5 5
1
b. log 4 2
c. log 81 27
d. log 398 1
e. log 275 0
f. log 8 −2
Properties of Logarithms
Property
Product
Definition
Quotient
Power
Equality
Ex 2: Solve the following using properties of logarithms
a. log 42 5𝑥 = log 42 7
b. log 6 2𝑥 + log 6 4𝑥 = log 6 64
c. log 3 (4𝑥 − 5) − log 3 (3 − 2𝑥) = 2
d. 3log 2 𝑥 + log 2 5 = log 2 125
Example
11.5 Common Logarithms
Common Log
Change of Base
Ex 1: Solve the following
a. 12𝑥 = 82
b. 63𝑥 = 81
c. 72𝑥+1 = 8𝑥−4
d. 2 log 𝑥 = 8
11.3 The Number “e”
Definition of Base “e”
Simple Exponential Growth
Compound Interest
Continuous Compound Interest
Ex 1: The rates for online savings accounts are as follows
Barclays
APR = 0.80%, compounded daily
Discover
APR = 1%, compounded daily
EverBank
APR = 0.90%, compounded monthly
Start with $10,000 and leave the money in the bank for 5 years. Find which account gives you the best return.
11.6 Natural Logarithms
Definition of Natural Log
Ex 1: Solve each equation. Round to 4 decimal places.
a. 5 + 𝑒 2𝑥 = 17
b. 2 + ln(2𝑥 + 3) = 11
c. 57𝑥−1 = 32𝑥+1
Radioactive Decay/Half-life
Ex 2: The half-life of Titanium 44 is 63 years. How long does it take for 39g of the substance to decay to 27g?
Ex 3: It takes 5 hrs. to clear half the amount of caffeine consumed out of an average person’s body. Sally had
a tall Starbucks coffee that contained 260 mg of caffeine. How long would it take to have 80% of the caffeine
to remain in her body? Round time to the nearest tenth of an hour.