Algebra 2 Warmup

9.2 Logarithms Start with an exponential function: y = bx When you flip that function over the y = x line, the function becomes x = by This is what we call the inverse. Given y = bx, y is called the logarithm of x y = logbx

Recap The logarithm of x with base b looks like this... y =logbx It is defined as the exponent y that makes the equation x = by true. So basically...y = logbx can be rewritten as x = by....they are the same just in different forms.

Example 1 Write each equation in exponential form. Remember: y = logbx x = by

Example 2 Write each equation in logarithmic form. Remember: x = by y = logbx

Example 3 Set the expression equal to y Rewrite in exponential form Evaluate the logarithmic expression Set the expression equal to y Rewrite in exponential form Write 64 with base of 2 Set exponents equal Solve Your Turn

Characteristics of a Logarithmic Function Function is continuous and one-to-one Domain is all positive real numbers Y-axis is an asymptote of the graph Range is all real numbers Graph contains the point (1,0)...x-intercept is 1.

Example 4 Evaluate each expression b) c) a) d) Notice a short cut?

Logarithmic Equation Equation that contains one or more logarithms. Example 5a & b: Solve the following equations. Remember: logbx = y x = by

Property of Equality for Logarithmic Functions If b is a positive number other than 1, then logbx = logby if and only if x = y. Example: If log7x = log73, then x = 3 The log bases must be equal!!

Logarithmic Inequality Change from logarithmic to exponential inequality (x can not be smaller than 0)

Solving log equations – Example 6 Extraneous solutions: Recall that the domain of a log function is all POSITIVE #’s...therefore you need to check solutions! CHECK!! Is undefined...solution is extraneous

Your turn – Example 7 CHECK!! Solve the equation and check your solution(s). CHECK!!

Homework Assignment #57 p. 515 23-34 all, 35-59 odd