Exponential and Logarithmic Functions 5

5.5 Exponential and Logarithmic Equations: Problem Solving EXPONENTIAL AND LOGARITHMIC FUNCTIONS Objectives Solve logarithmic equations. Use logarithms to solve application problems involving exponential growth and decay. Solve application problems involving Richter numbers. Use a change-of-base formula to evaluate a logarithm.

Exponential and Logarithmic Equations: Problem Solving Property 5.8 If x > 0, y > 0, b > 0, and b 1, then x = y if and only if log b x = log b y

Exponential and Logarithmic Equations: Problem Solving Solve 3 x = 5 to the nearest hundredth. Example 1

Exponential and Logarithmic Equations: Problem Solving Solution: By using common logarithms, we can proceed as follows: 3 x = 5 log 3 x = log 5 x log 3 = log 5 x = 1.46 nearest hundredth Example 1 Property 5.8 log r p = p log r

Exponential and Logarithmic Equations: Problem Solving Check: Because , we say that, to the nearest hundredth, the solution set for 3 x = 5 is {1.46}. Example 1

Logarithmic Equations Solve log x + log(x - 15) = 2. Example 4

Logarithmic Equations Solution: Because log 100 = 2, the given equation becomes log x + log(x - 15) = log 100 Now we can simplify the left side, apply Property 5.8, and proceed as follows: log [(x)(x – 15)] = log 100 x(x – 15) = 100 x 2 – 15x – 100 = 0 (x – 20)(x + 5) = 0 x – 20 = 0 or x + 5 = 0 x = 20 or x = -5 The domain of a logarithmic function must contain only positive numbers, so x and x - 15 must be positive in this problem. Therefore we discard the solution of -5, and the solution set is {20}. Example 4

Problem Solving How long will it take $500 to double if it is invested at 8% compounded quarterly? Problem 1

Problem Solving Solution: To double $500 means that the $500 will grow to $1000. We want to find out how long it will take; that is, what is t? Thus = 500( ) 4t = 500(1.02) 4t We multiply both sides of 1000 = 500(1.02) 4t by to get 2 = (1.02) 4t Therefore Problem 1

Problem Solving log 2 = (1.02) 4t = 4t log 1.02 Now let’s solve for t: 4t log 1.02 = log 2 t = 8.8 nearest tenth Therefore we are claiming that $500 invested at 8% interest compounded quarterly will double in approximately 8.8 years. Problem 1 Property 5.8 log r p = p log r

Problem Solving Check: $500 invested at 8% compounded quarterly for 8.8 years will produce = $500(1.02) 35.2 = $ Problem 1

Richter Numbers An earthquake that occurred in San Francisco in 1989 was reported to have a Richter number of 6.9. How did its intensity compare to the reference intensity? Problem 3

Richter Numbers Solution: I = ( )(I 0 ) I 7,943,282I 0 Thus its intensity was a little less than 8 million times the reference intensity. Problem 3

Logarithms with Base Other Than 10 or e Evaluate log Example 7

Logarithms with Base Other Than 10 or e Solution: Let x = log Changing to exponential form, we obtain 3 x = 41 Now we can apply Property 5.8. log 3 x = log 41 x log 3 = log 41 x = rounded to four decimal places Therefore we are claiming that 3 raised to the power will produce approximately 41. Check it! Example 7

Logarithms with Base Other Than 10 or e Using the method of Example 7 to evaluate log a r produces the following formula, which is often referred to as the change-of-base formula for logarithms.

Exponential and Logarithmic Equations: Problem Solving Property 5.9 If a, b, and r are positive numbers, with a 1 and b 1, then