1. MAT 171 Precalculus Algebra T rigsted - Pilot Test Dr. Claude Moore - Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions and Equations 5.1 Exponential Functions 5.2 The Natural Exponential Function 5.3 Logarithmic Functions 5.4 Properties of Logarithms 5.5 Exponential and Logarithmic Equations 5.6 Applications of Exponential and Logarithmic Functions

2. OBJECTIVES 1. Solving Exponential Equations 2. Solving Logarithmic Equations

3. Two Logarithmic properties log b u r = r log b u. Power Rule for logarithms If u = v, then log b u = log b v. Logarithm property of equality

4. To Solve Exponential Equations 1. If the equation can be written in the form b u = b v, then solve the equation u = v 2. If the equation cannot easily be written in the form b u = b v. a. Use the logarithm property of equality to “take the log of both sides” (typically using the base 10 or e). b. Use the product rule of logarithms to “bring down” any exponents. c. Solve for the given variable.

5. Solve each equation. Round to four decimals places. (3x - 2) (ln e) = ln 33

6. Properties of Logarithms If b > 0, b ≠ 1, u and v represent positive numbers and r is any real number, then Product rule: Quotient rule: Power rule: Logarithm Property of Equality If a logarithmic equation can be written in the form then u = v. Furthermore, if u = v, then

7. Solving Logarithmic Equations 1. Determine the Domain of the variable. 2. Use properties of logarithms to combine all logarithms, and write as a single logarithm, if needed. 3. Eliminate the logarithm by rewriting the equation in exponential form. 4. Solve for the given variable. 5. Check for any extraneous solutions. Verify that each solution is in the domain of the variable.

8. Domain is solution to x – 6 > 0 or x > 6 Domain is solution to 5x+6 > 0 and 3x-2 > 0 x > -5/6 and x > 2/3 Which is x > 2/3 Since 33/43 ≈ 0.767 is greater than 2/3 ≈ 0.667, x = 33/43 is the solution.

9. Domain is solution to x – 2 > 0 or x > 2.

11. 14. 2(e x-1 ) 2 e 3-x = 80 2(e 2x-2 ) e 3-x = 80 e (2x-2)+(3-x) = 40 e x+1 = 40 ln e x+1 = ln 40 x+1 = ln 40 x = -1 + ln 40 x ≈ 2.6889 13. e x-3 e 3x+7 = 24 e (x-3)+(3x+7) = 24 e 4x+4 = 24 ln e 4x+4 = ln 24 4x+4 = ln 24 4x = -4 + ln 24 x = 0.25(-4 + ln 24) x ≈ -0.2055

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