Logarithmic Functions and Models Lesson 5.4

A New Function Consider the exponential function y = 10x Based on that function, declare a new function x = log10y You should be able to see that these are inverse functions In general The log of a number is an exponent

Note: if no base specified, default is base of 10 The Log Function Try These log39 = ? log232 = ? log 0.01 = ? Note: if no base specified, default is base of 10

Graph, Domain, Range Use your calculator to discover facts about the log function In the Y= screen, specify log(x) Set tables with T initial x = 0, x = 0.1 View the tables

Graph, Domain, Range Note domain for 0 < x < 1 Change the x to 5, view again

Graph, Domain, Range View graph with window -1 < x < 10, -4 < y < 5 Why does the graph appear undefined for x < 0 ?

Graph, Domain, Range Recall that There can be no value for y that gives x < 0 Domain for y = log x x > 0 Range y = { all real values }

Vertical Asymptote Note behavior of function as x  0+

Inverse Properties Explain why the following would be true. Note the graphical relationship of y = 10x y = log x and

Solving Exponential Equations Consider Divide by 2 Take log both sides Rewrite using inverse

Try It Out Consider solution of Steps Isolate the 10x Take log of both sides Use the inverse property

Modeling Data with Logarithms Consider the table below We seek to model this data with a function Substitute values to get two equations with a and b – solve the equations Acres 10 100 1000 100000 Types of Insects 500 800 1100 1400 1700

Modeling Data with Logarithms Substitute values of x and y Now use substitution for a and b Finally f(x) = 200 + 300 log x

Logarithmic Equations Consider solving the logarithmic equation log 4x = 2 Exponentiate both sides using the base Use the inverse property … and solve

Assignment Lesson 5.4A Page 402 Exercises 1 – 71 EOO Lesson 5.4B Exercises 75 – 89 odd