Pre-AP Pre-Calculus Chapter 3, Section 4 Properties of Logarithmic Functions 2013 - 2014

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

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

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.

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.

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

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 2 5 + 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

Change of Base When working with logs, sometimes you get bases and numbers that don’t go together like log 4 7 . 7 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 4 7 .

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

Evaluating Logarithms by Changing the Base

Ch. 3.4 Homework Pg. 317, #’s: 1 – 35 every other odd, 39, 41, 43 13 total problems Gray Book: pg. 289 - 291