Exponential and Logarithmic Equations and Inequalities 7 5

Exponential and Logarithmic Equations and Inequalities 7 -5 LEARNING GOALS – LESSON 7. 5. 1: Solve exponential and logarithmic equations and equalities. 7. 5. 2: Solve problems involving exponential and logarithmic equations. To solve exponential equations: • Try writing them so that the bases are all the same. • Take the logarithm of both sides. A. Example 1 A: Solving Exponential Equations Solve. 9 8–x = 27 x – 3 Rewrite each side with the same base; 9 and 27 are powers of 3. To raise a power to a power, multiply exponents. DON’T FORGET TO _____ Bases are the same, so the exponents must be equal. Solve for x. B. Example 1 B: Solving Exponential Equations Solve. 4 x– 1 = 5 5 is not a power of 4, so take the log of both sides. Apply the Power Property of Logarithms. Divide both sides by log 4.

Exponential and Logarithmic Equations and Inequalities 7 -5 C. Solve. 32 x = 27 Rewrite each side with the same base; 3 and 27 are powers of 3. To raise a power to a power, multiply exponents. Bases are the same, so the exponents must be equal. Solve for x. D. Solve. 7 –x = 21 21 is not a power of 7, so take the log of both sides. Apply the Power Property of Logarithms. Divide both sides by log 7. E. Solve. 23 x = 15 15 is not a power of 2, so take the log of both sides. Apply the Power Property of Logarithms. Divide both sides by log 2, then divide both sides by 3.

7 -5 Exponential and Logarithmic Equations and Inequalities Example 2: Financial Application A. You receive one penny on the first day, and then triple that (3 cents) on the second day, and so on for a month. On what day would you receive a least a million dollars. Take the log of both sides. Use the Power of Logarithms. log 108 is 8. Divide both sides by log 3. Evaluate by using a calculator. Round up to the next whole number. B. A single cell divides every 5 minutes. How long will it take for one cell to become more than 10, 000 cells?

Exponential and Logarithmic Equations and Inequalities 7 -5 You can solve logarithmic equations by using the properties of logs. Example 3: Solving Logarithmic Equations A. Solve. log 6(2 x – 1) = – 1 Use 6 as the base for both sides. Use inverse properties to remove 6 to the log base 6. Simplify. B. Solve. log 4100 – log 4(x + 1) = 1 Write as a quotient. Use 4 as the base for both sides. Use inverse properties on the left side. C. Solve. log 5 x 4 = 8 Power Property of Logarithms. Divide both sides by 4 to isolate log x. 5 Definition of a logarithm.

Exponential and Logarithmic Equations and Inequalities 7 -5 D. Solve. log x + log 12(x + 1) = 1 12 Product Property of Logarithms. Exponential form. Use the inverse properties. Multiply and collect terms. Factor. Set each of the factors equal to zero. Solve. Caution Watch out for calculated solutions that are not solutions of the original equation. Example 4: Solving Exponential Equations w/ a Calculator Use a table and graph to solve the equation 23 x = 33 x– 1.