Chapter 5 Exponential Functions and Logarithmic Functions Copyright

  • Slides: 14
Download presentation
Chapter 5 Exponential Functions and Logarithmic Functions Copyright © 2016, 2012 Pearson Education, Inc.

Chapter 5 Exponential Functions and Logarithmic Functions Copyright © 2016, 2012 Pearson Education, Inc. 5 -1

Section 5. 1 5. 2 5. 3 5. 4 5. 5 Inverse Functions Exponential

Section 5. 1 5. 2 5. 3 5. 4 5. 5 Inverse Functions Exponential Functions and Graphs Logarithmic Functions and Graphs Properties of Logarithmic Functions Solving Exponential Equations and Logarithmic Equations 5. 6 Applications and Models: Growth and Decay; and Compound Interest Copyright © 2016, 2012 Pearson Education, Inc. 5 -2

5. 5 Solving Exponential and Logarithmic Equations · · Solve exponential equations. Solve logarithmic

5. 5 Solving Exponential and Logarithmic Equations · · Solve exponential equations. Solve logarithmic equations. Copyright © 2016, 2012 Pearson Education, Inc. 5 -3

Solving Exponential Equations with variables in the exponents, such as 3 x = 20

Solving Exponential Equations with variables in the exponents, such as 3 x = 20 and 25 x = 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 © 2016, 2012 Pearson Education, Inc. 5 -4

Example Solve Solution: Write each side as a power of the same number (base).

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: The solution is 4. Copyright © 2016, 2012 Pearson Education, Inc. TRUE 5 -5

Another Property of Logarithmic Equality For any M > 0, N > 0, and

Another Property of Logarithmic Equality For any M > 0, N > 0, and a 1, loga M = loga N M = N. Copyright © 2016, 2012 Pearson Education, Inc. 5 -6

Example Solve: 3 x = 20. Solution: This is an exact answer. We cannot

Example Solve: 3 x = 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. The solution is about 2. 7268. Copyright © 2016, 2012 Pearson Education, Inc. 5 -7

Example Solve: e 0. 08 t = 2500. Solution: The solution is about 97.

Example Solve: e 0. 08 t = 2500. Solution: The solution is about 97. 8. Copyright © 2016, 2012 Pearson Education, Inc. 5 -8

Example Solve: 4 x+3 = 3 -x. Solution: Copyright © 2016, 2012 Pearson Education,

Example Solve: 4 x+3 = 3 -x. Solution: Copyright © 2016, 2012 Pearson Education, Inc. 5 -9

Solving Logarithmic Equations containing variables in logarithmic expressions, such as log 2 x =

Solving Logarithmic Equations containing variables in logarithmic expressions, such as log 2 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 © 2016, 2012 Pearson Education, Inc. 5 -10

Example Solve: log 3 x = 2. Check: Solution: TRUE The solution is Copyright

Example Solve: log 3 x = 2. Check: Solution: TRUE The solution is Copyright © 2016, 2012 Pearson Education, Inc. 5 -11

Example Solve: Solution: Copyright © 2016, 2012 Pearson Education, Inc. 5 -12

Example Solve: Solution: Copyright © 2016, 2012 Pearson Education, Inc. 5 -12

Example (continued) Check x = 2: q Check x = – 5: FALSE TRUE

Example (continued) Check x = 2: q 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 © 2016, 2012 Pearson Education, Inc. 5 -13

Example Solve: Solution: Only the value 2 checks and it is the only solution.

Example Solve: Solution: Only the value 2 checks and it is the only solution. Copyright © 2016, 2012 Pearson Education, Inc. 5 -14