Download Lesson 4: Properties of Logarithms

Algebra IIA
Unit IV: Exponential Functions and Logarithms
 Foundational Material
o Use the properties of exponents to simplify
expressions
o Perform inverse operations
o Solve problems and fit data using linear and
quadratic and polynomial models
 Goal
o Study exponential functions
o Study logarithms, the inverse
o Solve problems and fit data
and use properties of
of exponents and logarithmic
to exponential and
exponents
functions
logarithmic models
 Why?
o To further build a foundation for higher level mathematics such as statistics and business calculus
o These skills can be used to observe, understand, and model relationships in science, social studies and economics
o Maximize and minimize area and volume

o
o
o
o
o
Key Vocabulary
Asymptote
Base
Change of Base Formula
Common Logarithm
Exponential Decay
o
o
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o
o
Exponential Growth
Exponential Equation
Exponential Regression
Inverse Function
Logarithmic Equation
o
o
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Logarithmic Function
Logarithmic Regression
Natural Logarithm
Properties of Logarithms
Lesson 4: Properties of Logarithms


Use properties to simplify logarithmic expressions.
Translate between logarithms in any base.
Properties of Logarithms
1. log b 1  0
2. log b b  1
3. log b b x  x
4. b logb x  x
5. If b x  b y , then x  y
More Properties of Logarithms
Logarithm of a Product
logb (xy) = logb x + logb y
Example #1: Rewrite as a sum.
log(21) =
Logarithm of a Quotient
x
logb ( ) = logb x – logb y
y
Example #2: Rewrite as a difference.
y
log3 =
8
Logarithm of a Power
logb (xn) = n logb x
Example #3: Rewrite as a ‘linear’ log expression.
loga 3 x =
Change of Base Formula
log b
log a b =
log a
Examples: Simplify each expression.
Example #4: Evaluate the logarithm.
log5 23 =
a. log3 3x
c. log3 81x
b. 5log5 3x
d. log7 ( 2x +1) = log7 5
e. log8y = log5
Graph the following.
1. y = log3x
2. y = log5x
Examples:
1. Without a calculator! Find log 15 given
log 3  .47712 and log 5  .69897
2. Without a calculator! Find log 48 given
log 2  .301 and log 3  .477
3. Express the following as a single logarithm:
5log2m – log2p
log7 3 + 2 log7 5
log5x -
1
log5y
5
4. Simplify each expression completely.
log232 – log242
3
5. Evaluate the logarithmic expressions.
log6 39 =
1
3 9
log 1 27 + log 1
æ1ö
log 2 ç ÷
è2ø
5
log7 49 - log7 7
log3 95 =
Assignment: Page 260, 21-35 odd, 41-45 odd, 66-68