Unit 5: Modeling with Exponential & Logarithmic FunctionsMs. C. Taylor

Warm-Up

Identify the value of b in the following:

Graphing Exponential Equations

The graph will approach the axis but will never touch.

Asymptote for the function will approach the x-axis.

Asymptote for the inverse function will approach the y-axis.

Warm-Up

Rewrite using exponent rules

Logarithms

Suppose b>0 and b≠1. For x>0, there is a number y such that if and only if

LogarithmicExponential Form

ExponentialLogarithmic Form

Inverse Property of Exponents & Logarithms

LogarithmicExponential Inequality

If

If

Property of Equality for Logarithmic Functions

If b is a positive number other than 1, then if and only if

Example: If , then

Property of Inequality for Logarithmic Functions

If , then if and only if, and if and only if

If , then

Product Property of Logarithms

For all positive numbers m, n, and b, where b≠1,

Example #1

Expand the following logarithms:

Example #2

Use to approximate the value of Use to approximate the value of

Quotient Property of Logarithms

For all positive numbers m, n, and b, where ,

Example #3

Expand the following logarithms:

Example #4

Use and to approximate Use and to approximate

Power Property of Logarithms

For any real number p and positive numbers m and b, where ,

Examples

Given , approximate the value of

Given , approximate the value of

Warm-Up

Expand the following:

Find Common Logarithms

Change of Base Formula

For all positive numbers, a, b, and n, where and ,

ExamplesExpress in terms of common

logarithms. Then approximate its value to four decimal places.

Express in terms of common logarithms. Then approximate its value to four decimal places.

Evaluate Natural Base Expressions

Evaluate Natural Logarithmic Expressions

Equivalent Expressions

If something has an e in it then that will become a ln.

If something has an ln in it then it will become e raised to a power.

Warm-UpEvaluate the following

Warm-Up

Use the properties of logarithms to rewrite:

Inverse Property of Base e & Natural

Logarithms

Evaluate Logarithmic Expressions

Solve Logarithmic Equations

Solve Equations with Logarithms on Both

Sides

Solve

Solve Equations using Properties of Logarithms

Warm-Up

log 𝑥− log (𝑥−1 )=log (3 𝑥+12)

Solve Exponential Equations using

Logarithms

Solve Base e Equations

Solve Natural Log Equations & Inequalities