CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest Copyright © 2009 Pearson Education, Inc.

5.5 Solving Exponential and Logarithmic Equations Solve exponential equations. Solve logarithmic equations. Copyright © 2009 Pearson Education, Inc.

Solving Exponential Equations Equations with variables in the exponents, such as 3x = 20 and 25x = 64, are called exponential equations. Use the following property to solve exponential equations. Base-Exponent Property For any a > 0, a  1, ax = ay  x = y. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve Solution: Write each side as a power of the same number (base). Since the bases are the same number, 2, we can use the base-exponent property and set the exponents equal: Check x = 4: TRUE The solution is 4. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Another Property Property of Logarithmic Equality For any M > 0, N > 0, a > 0, and a  1, loga M = loga N  M = N. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: 3x = 20. Solution: This is an exact answer. We cannot simplify further, but we can approximate using a calculator. We can check by finding 32.7268  20. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: e0.08t = 2500. Solution: The solution is about 97.8. Copyright © 2009 Pearson Education, Inc.

Solving Logarithmic Equations Equations containing variables in logarithmic expressions, such as log2 x = 4 and log x + log (x + 3) = 1, are called logarithmic equations. To solve logarithmic equations algebraically, we first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: log3 x = 2. Check: Solution: TRUE The solution is Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: Solution: Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example (continued) Check x = 2: Check x = –5: FALSE TRUE The number –5 is not a solution because negative numbers do not have real number logarithms. The solution is 2. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: Solution: Only the value 2 checks and it is the only solution. Copyright © 2009 Pearson Education, Inc.

Example - Using the Graphing Calculator Solve: e0.5x – 7.3 = 2.08x + 6.2. Solve: Graph y1 = e0.5x – 7.3 and y2 = 2.08x + 6.2 and use the Intersect method. The approximate solutions are –6.471 and 6.610. Copyright © 2009 Pearson Education, Inc.