MAT 150 – Class #17 Topics: Graph and evaluate Logarithmic Functions Convert equations to Logarithmic and Exponential Forms Evaluate and Apply Common Logarithms Evaluate and Apply Natural Logarithms Apply Logarithmic Properties.

Graphing Graph each of the following functions using a table of values. (Hint: The log button on your calculator assumes that it is with a base of 10) 𝑦= 10 𝑥 𝑦= log 10 𝑥 What do you notice about the two graphs? Therefore, exponential functions and logarithmic functions are inverses (opposites) of each other.

Writing Logarithms in Exponential Form Logarithmic 25 = 5² 729= 3 6 𝑦= 𝑏 𝑥 log 𝑏 𝑦 =𝑥

Some Properties of Logarithms Property Example log 𝑏 𝑏 =1 log 𝑏 1 =0 log 𝑏 ( 𝑀𝑁 )= log 𝑏 𝑀+ log 𝑏 𝑁 log 𝑏 𝑀 𝑁 = log 𝑏 𝑀 − log 𝑏 𝑁 log 𝑏 𝑀 𝑥 =𝑥∙ log 𝑏 𝑀

Expanding Logarithms Rewrite each of the following expressions as a sum, difference or product of logarithms and simplify if possible. log 4 5(𝑥−7) ln [ 𝑒 2 𝑒+3 ] log 𝑥 −8 𝑥 log 4 𝑦 6 ln 1 𝑥 5

Single Logarithms Rewrite each of the following expressions as a single logarithm. log 3 𝑥 +4 log 3 𝑦 = 1 2 log 𝑎 −3 log 𝑏 = ln 5𝑥 −3 ln 𝑧 =

Japan’s Population The population of Japan for the years 1984 – 2006 is approximated by the logarithmic function 𝑦=114.016+4.4.267 ln 𝑥 million people, with x equal to the number of years after 1980. According to the model, what is the estimated population in 2000? In 2020?

Assignment Pg. 337-340 #1-11 odd #15-17 odd #31-32 #35-37 odd #39, 41 REMINDER: TEST 3 DUE AT THE BEGINNING OF CLASS FRIDAY