10.2 Logarithms & Logarithmic Functions Objectives: The students will be able to… Evaluate logarithmic expressions Solve logarithmic equations and inequalities.

Recall the graph of What does its inverse look like?

The inverse of can be defined as In general, the inverse of is In , y is called the logarithm of x. It is usually written as and is read “log base b of x”.

Logarithm with Base b Let b and x be positive numbers, The logarithm of x with base b is denoted It is defined as the exponent y that makes the equation true.

Example 1: Logarithmic to Exponential Form Write each equation in exponential form:

Example 2: Exponential to Logarithmic Form Write each equation in logarithmic form:

You Try It… State whether each equation is in exponential or logarithmic form, then switch forms. b) c) d)

Example 3: Evaluate Logarithmic Expressions: a) b)

Evaluate each expression. You Try It… Evaluate each expression. b) d) e)

Properties of Logarithmic Functions Logarithmic functions are the INVERSE of exponential functions The function is continuous and one-to-one. The domain is the set of all positive real numbers. The y-axis is an asymptote of the graph. The range of all real numbers. The graph contains the point (1,0). That is, the x-intercept is 1.

What is always true about INVERSE functions?

Example 4: Inverse Property of Exponents and Logarithms Evaluate each expression:

Example 5: Solve a Logarithmic Equation

You Try It… Solve the logarithmic equation: a) b)

Property of Equality for Logarithmic Functions If b is a positive number other than 1, then logbx = logby IFF (if and only if) x = y Example: If log7x = log73, then x = 3

Example 7: Solve Equations with Logarithms on Each Side

Solve each equation. Check your solutions. You Try It… Solve each equation. Check your solutions.