1. WARM UP Simplify 1. 2.

2. USING A CALCULATOR Use a calculator or Table 2 1.Find log 2.13 2.Find log 432 3.Find antilog 0.3263 4.Find antilog -0.4413 0.3284 2.6355 2.12 0.362

3. EXPONENTIAL AND LOGARITHMIC EQUATIONS

4. OBJECTIVES  Solve exponential equations  Solve logarithmic equations  Solve problems involving exponential and logarithmic equations.  Solve real-world problems involving exponential and logarithmic relationships.

5. EXPONENTIAL EQUATIONS  Earthquake intensity, loudness of sound, and compound interest are all applications of exponential and logarithmic equations.  An equation with variables in exponents such as is called an exponential equation.  We can solve such equations by taking logarithms of both sides and then using Theorem 12-5 using the form:

6. EXAMPLE 1 Solve 3 = 8 Taking the log of both sides (Remember log m = log m) Using Theorem 12-5 Solving for x We look up the logs, or find them on a calculator and divide.

7. EXAMPLE 2 Solve 2 = 16 Taking the log of both sides Using Theorem 12-5 Solving for x and evaluating logarithms Calculating The answer is approximate because the logarithms are approximate. We can see that 3 is the solution since

8. EXAMPLE 3  The following is another method of solving exponential equations. Solve 2 = 16 Note that 16 = 2. We have Since the base, 2, is the same on both sides, the exponents must be equal. 3x – 5 = 4, or x = 3

9. TRY THIS… a.Solve 2 = 7 b.Solve 4 = 6 c.Solve 4 = 65. Use the method in Example 6

10. LOGARITHMIC EQUATIONS  Equations that contain logarithmic expressions are logarithmic equations..  We solve them by converting to an equivalent exponential equation. For example, to solve log x = -3, we convert to x = 2 and find that x =  To solve logarithmic equations we first try to obtain a single logarithmic expression on one side of the equation and then write an equivalent exponential equation.

11. EXAMPLE 4 Solve We already have a single logarithmic expression, so we write an equivalent exponential equation. Check:

12. EXAMPLE 5 Possible solutions to logarithmic equations must be checked because domains of logarithmic functions consist only of positive numbers. Check: The number -2 is not a solution because negative numbers do not have logarithms. The solution is 5.

13. TRY THIS… Solve 1. 2. 3.

14. PROBLEM SOLVING: LOGARITHMS  The amount A that principal P will be worth after t years at interest rate 4, compounded annually, is given by the formula A = P(1 + r) Example 6: Suppose $4,000 principal is invested at 6% interest and yields $5353. For how many years was it invested. We use the formula A = P(1 + r) 5,353 = 4,000(1 + 0.06) or 5353 = 4000(1.06) Then we solve for t. log 5353 = log (4000(1.06) ) Taking the log of both sides log 5353 = log 4000+ t log 1.06 Using Theorems 12-4 and 12-5 Solving for t Evaluating logarithms The money was invested for approximately 5 years. We can use a calculator to check. 4000(1.06) ≈ 5353.9023. The solution checks.

15. TRY THIS… Suppose $5000 was invested at 14%, compounded annually, and it yielded $18,540. For how long was it invested?

16. LOGARITHMIC PROBLEMS  The sensation of loudness of sound is not proportional to the energy intensity but rather is a logarithmic function.  Loudness is measured in bels (after Alexander Graham Bell) or in smaller units, decibels. Loudness in decibels of a sound of intensity (I) is defined to be  where I is the minimum intensity detectable by the human ear (such as the tick of a watch at 6 meters under quiet conditions). When a sound is 10 times as intense as another, it is louder by 20 decibels, and so on.

17. EXAMPLE 7 a. Find the loudness in decibels of the background noise in a radio station for which the intensity (I) is 199 times I We substitute into the formula and calculate, using a calculator. b. Find the loudness of the sound of a rock concert, for which the intensity is 10 times I 10 199 = 22.98853076 ≈ 23 decibels X LOG = 110 decibels

18. TRY THIS… a.Find the loudness in decibels of the sound in a library, for which the intensity is 2510 times I. b.Find the loudness in decibels of conversational speech, for which the intensity is 10 times I.

19. EXAMPLE 8  The magnitude R (on the Richter scale) of an earthquake of intensity I is defined as where I is a minimum intensity used for comparison. An earthquake has an intensity 4 X 10 times I. What is its magnitude on the Richter scale? We substitute into the formula. 4 8 8.60205991 ≈ 8.6 LOG+ = The magnitude on the Richter Scale is about 8.6

20. TRY THIS… a.The earthquake in Anchorage, Alaska on March 27, 1964 had an intensity of 2.5 X 10 times I. What was its magnitude on the Richter scale?

21. CH. 12.7 HOMEWORK SOLVING EXPONENTIAL & LOGARITHM EQUATIONS: Textbook pg. 547 #2, 6, 10, 14, 16, 20 24, 26 & 28