WarmUp 10 2 Logarithmic Functions Find the inverse

Warm-Up 10. 2 Logarithmic Functions Find the inverse of each function. 1) f(x) = x + 10 2) g(x) = 3 x 3) h(x) = 5 x + 3 4) j(x) = ¼x + 2

10. 2 Logarithmic Functions • Write equivalent forms for exponential and logarithmic equations. • Use the definitions of exponential and logarithmic functions to solve equations.

10. 2 Logarithmic Functions Rules and Properties Equivalent Exponential and Logarithmic Forms For any positive base b, where b 1: bx = y if and only if x = logb y. Exponential form Logarithmic form

Example 1 6. 3 Logarithmic Functions a) Write 27 = 128 in logarithmic form. log 2 128 = 7 b) Write log 6 1296 = 4 in exponential form. 64 = 1296

Example 2 6. 3 Logarithmic Functions a. Solve x = log 2 8 for x. 2 x = 8 x= 3 b. logx 25 = 2 x 2 = 25 x= 5

Practice 6. 3 Logarithmic Functions c. Solve log 2 x = 4 for x. 24 = x x = 16

6. 3 Logarithmic Functions Example 3 a. Solve 10 x = 14. 5 for x. Round your answer to the nearest tenth. log 1014. 5 = x x = 1. 161

Rules and Properties 6. 3 Logarithmic Functions One-to-One Property of Exponential Functions If bx = by, then x = y.

6. 3 Logarithmic Functions Example 4 Find the value of the variable in each equation: a) log 2 1 = r 2 r = 1 20 = 1 r=0 b) log 7 D= 3 73 = D D = 343

6. 3 Logarithmic Functions Practice Find the value of the variable in each equation: 1) log 4 64 = v 2) logv 25 = 2 3) 6 = log 3 v

Practice Solve each equation for x. Round your answers to the nearest hundredth. 1) 10 x = 1. 498 2) 10 x = 0. 0054 Find the value of x in each equation. 3) x = log 4 1 4) ½ = log 9 x

Properties of Logarithmic Functions Objectives: • Simplify and evaluate expressions involving logarithms • Solve equations involving logarithms

Properties of Logarithms For m > 0, n > 0, b > 0, and b 1: Product Property logb (mn) = logb m + logb n

Example 1 given: log 5 12 1. 5440 log 5 10 1. 4307 log 5 120 = log 5 (12)(10) = log 5 12 + log 5 10 1. 5440 + 1. 4307 2. 9747

Properties of Logarithms For m > 0, n > 0, b > 0, and b 1: Quotient Property m logb = logb m – logb n n

Example 2 given: log 5 12 1. 5440 log 5 10 1. 4307 12 log 5 1. 2 = log 5 10 = log 5 12 – log 5 10 1. 5440 – 1. 4307 0. 1133

Properties of Logarithms For m > 0, n > 0, b > 0, and any real number p: Power Property logb mp = p logb m

Example 3 given: log 5 12 1. 5440 log 5 10 1. 4307 log 5 1254 = 4 log 5 125 = 4 3 = 12 5 x = 125 53 = 125 x=3

Practice Write each expression as a single logarithm. 1) log 2 14 – log 2 7 2) log 3 x + log 3 4 – log 3 2 3) 7 log 3 y – 4 log 3 x

4 minutes Warm-Up Write each expression as a single logarithm. Then simplify, if possible. 1) log 6 6 + log 6 30 – log 6 5 2) log 6 5 x + 3(log 6 x – log 6 y)

Properties of Logarithms For b > 0 and b 1: Exponential-Logarithmic Inverse Property logb bx = x and b logbx = x for x > 0

Example 1 Evaluate each expression. a) b)

Practice Evaluate each expression. 1) 7 log 711 – log 3 81 2) log 8 85 + 3 log 38

Properties of Logarithms For b > 0 and b 1: One-to-One Property of Logarithms If logb x = logb y, then x = y

Example 2 Solve log 2(2 x 2 + 8 x – 11) = log 2(2 x + 9) for x. log 2(2 x 2 + 8 x – 11) = log 2(2 x + 9) 2 x 2 + 8 x – 11 = 2 x + 9 2 x 2 + 6 x – 20 = 0 2(x 2 + 3 x – 10) = 0 2(x – 2)(x + 5) = 0 x = -5, 2 Check: log 2(2 x 2 + 8 x – 11) = log 2(2 x + 9) log 2 (– 1) = log 2 (-1) undefined log 2 13 = log 2 13 true

Solve for x. Practice 1) log 5 (3 x 2 – 1) = log 5 2 x 2) logb (x 2 – 2) + 2 logb 6 = logb 6 x

Exponential Growth and Decay Objectives: • Determine the multiplier for exponential growth and decay • Write and evaluate exponential expressions to model growth and decay situations

Modeling Bacteria Growth Time (hr) 0 Population 25 1 2 3 4 5 6 50 100 200 400 800 1600 Write an algebraic expression that represents the population of bacteria after n hours. The expression is called an exponential expression because the exponent, n is a variable and the base, 2, is a fixed number. The base of an exponential expression is commonly referred to as the multiplier.

Example 1 Find the multiplier for each rate of exponential growth or decay. a) 9% growth 100% + 9% = 109% = 1. 09 b) 0. 08% growth 100% + 0. 08% = 100. 08% = 1. 0008 c) 2% decay 100% - 2% = 98% = 0. 98 d) 8. 2% decay 100% - 8. 2% = 91. 8% = 0. 918

Example 2 Suppose that you invested $1000 in a company’s stock at the end of 1999 and that the value of the stock increased at a rate of about 15% per year. Predict the value of the stock, to the nearest cent, at the end of the years 2004 and 2009. Since the value of the stock is increasing at a rate of 15%, the multiplier will be 115%, or 1. 15 = $2011. 36 = $4045. 56

Example 3 Suppose that you buy a car for $15, 000 and that its value decreases at a rate of about 8% per year. Predict the value of the car after 4 years and after 7 years. Since the value of the car is decreasing at a rate of 8%, the multiplier will be 92%, or 0. 92 = $10, 745. 89 = $8, 367. 70

Practice A vitamin is eliminated from the bloodstream at a rate of about 20% per hour. The vitamin reaches a peak level in the bloodstream of 300 mg. Predict the amount, to the nearest tenth of a milligram, of the vitamin remaining 2 hours after the peak level and 7 hours after the peak level.