1 6.6 Logarithmic and Exponential Equations In this section, we will study the following topics: Solving logarithmic equations Solving exponential equations Using exponential and logarithmic equations to solve real-life problems.

2 Strategies for Solving Logarithmic Equations There are two basic strategies for solving logarithmic equations— 1. Converting the log equation into an exponential equation by using the definition of a logarithm: 2. Using the One-to-One Property:

3 Solving Logarithmic Equations I. Converting to Exponential Form 1. ISOLATE the logarithmic expression on one side of the equation. 2. CONVERT TO EXPONENTIAL FORM 3. SOLVE for x. Give approximate answers to 3 decimal places, unless otherwise indicated.

4 Example: Converting to Exponential Form Solve

5 Solving Logarithmic Equations II. Using the One-to-One* Properties  *Can be used if the equation can be written so that both sides are expressed as SINGLE LOGARITHMS with the SAME BASE. 1.Use the properties of logarithms to CONDENSE the log expressions on either side of the equation into SINGLE LOG expressions. 2.Apply the ONE-TO-ONE PROPERTY. 3.SOLVE for x. Give approximate answers to 3 decimal places, unless otherwise indicated.

6 Example: One-to-One Property for Logs Solve log 3 x + 2log 3 5 = log 3 (x + 8)

7 Strategies for Solving Exponential Equations There are two basic strategies for solving exponential equations— 1. Using the One-to-One Property: 2. Taking the natural or common log of each side.

8 Solving Exponential Equations I. Using the One-to-One property *Can be used if the equation can be written so that both sides are expressed as powers of the SAME BASE Use the properties of exponents to CONDENSE the exponential expressions on either side of the equation into SINGLE exponential expressions. Apply the ONE-TO-ONE PROPERTY. SOLVE for x. Give approximate answers to 3 decimal places, unless otherwise indicated.

9 Example: One-to-One Property Solve

10 Example: One-to-One Property Solve

11 II. Taking the Natural or Common Log of Each Side If there is one exponential term, ISOLATE the exponential term on one side of the equation. TAKE THE NATURAL OR COMMON LOG OF EACH SIDE of the equation. Use the property to get the variable out of the exponent. SOLVE for x. Give approximate answers to 3 decimal places, unless otherwise indicated. Solving Exponential Equations

12 Solve 3(5 4x+1 ) -7 = 10 Example: Taking the Log of Each Side

13 Solve. Example: Taking the Log of Each Side

14 Solving Exponential and Logarithmic Equations Graphically Remember, you can verify the solution of any one of these equations by finding the graphical solution using your TI-83/84 calculator. Enter the left hand side of the original equation in y 1 Enter the right side in y 2 Find the point at which the graphs intersect. Below is the graphical solution of the equation 2ln(x - 5) = 6 The x-coordinate of the intersection point is approximately This is the (approx) solution of the equation.

15 Solve each of the following EXPONENTIAL equations. PRACTICE!!

16 PRACTICE!!

17 Solve each of the following LOGARITHMIC equations. PRACTICE!!

18 PRACTICE!!

19 PRACTICE!!

20 A Compound Interest Example How long will it take $25,000 to grow to $500,000 at 9% annual interest compounded continuously? Use the compound interest formula: Where P = Principal (original amount invested or borrowed) r = annual interest rate t = number of years money is invested A = the amount of the investment after t years

21 A Compound Interest Example (cont.) Substitute in the given values and solve for t.

22 The population of Asymptopia was 6500 in 1970 and has been tripling every 12 years since then. When will the population reach 75,000? Let t represent the number of years since 1970 P(t) represents the population after t years. Another Example

23  End of Section 6.6