1. Pre-AP Pre-Calculus Chapter 3, Section 4
Properties of Logarithmic Functions

2. Properties of Logarithms
Let b, R, and S be positive real numbers with b ≠ 1, and c any real number.
Product Rule: log 𝑏 𝑅𝑆 = log 𝑏 𝑅 + log 𝑏 𝑆
Quotient Rule: log 𝑏 𝑅 𝑆 = log 𝑏 𝑅 − log 𝑏 𝑆
Power Rule: log 𝑏 𝑅 𝑐 =𝑐 log 𝑏 𝑅

3. Exploration 1: Exploring the Arithmetic of Logarithms
Use the 5-decimal place approximation shown to support the properties of logarithms numerically.
Product log 2∙4
Quotient log 8 2
Power log 2 3

4. Expanding the Logarithm of a Product
Assuming x and y are positive, use properties of logarithms to write log 8𝑥 𝑦 4 as a sum of logarithms or multiples of logarithms.

5. Expanding the Logarithm of a Product
Assuming x and y are positive, use properties of logarithms to write ln 𝑥 2 +5 /𝑥 as a sum of logarithms or multiples of logarithms.

6. Condensing a Logarithmic Expression
Assyming x and y are positive, write ln 𝑥 5 −2 ln (𝑥𝑦) as a single logarithm.

7. Discovering Relationships and Nonrelationships
Of the eight relationships suggested here, four are true and four are false. Determine which ones are the true functions and which ones are false.
1. ln (𝑥+2) = ln 𝑥 + ln 2
2. log 3 (7𝑥) =7 log 3 𝑥
3. log 𝑥 5𝑥 = log log 2 𝑥
4. ln 𝑥 5 = ln 𝑥− ln 5
5. log 𝑥 4 = log 𝑥 log 4
6. log 4 𝑥 3 =3 log 4 𝑥
7. log 5 𝑥 2 =( log 5 𝑥) ( log 5 𝑥 )
8. log 4𝑥 = log 4 + log 𝑥

8. Change of Base
When working with logs, sometimes you get bases and numbers that don’t go together like log is not a simple power of 4 and there is no way to compute log base 4 on the calculator. You can use some algebra to make this work. First let 𝑦= log

9. Change-of-base Formula for Logarithms
For positive real numbers a, b, and x with a ≠ 1 and b ≠ 1,
log 𝑏 𝑥 = log 𝑎 𝑥 log 𝑎 𝑏
Since calculators have two logarithm keys which correspond to the bases 10 and e. We can use one of the following forms:
log 𝑏 𝑥 = log 𝑥 log 𝑏 𝑜𝑟 log 𝑏 𝑥 = ln 𝑥 ln 𝑏

10. Evaluating Logarithms by Changing the Base

11. Ch. 3.4 Homework Pg. 317, #’s: 1 – 35 every other odd, 39, 41, 43
13 total problems
Gray Book: pg