Warm-Up Find the inverse of each function. f(x) = x + 10 g(x) = 3x 10.2 Logarithmic Functions Find the inverse of each function. f(x) = x + 10 g(x) = 3x h(x) = 5x + 3 j(x) = ¼x + 2

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

Rules and Properties Equivalent Exponential and Logarithmic Forms 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 a) Write 27 = 128 in logarithmic form. log2 128 = 7 6.3 Logarithmic Functions a) Write 27 = 128 in logarithmic form. log2 128 = 7 b) Write log6 1296 = 4 in exponential form. 64 = 1296

Example 2 a. Solve x = log2 8 for x. 2x = 8 x = 3 b. logx 25 = 2 6.3 Logarithmic Functions a. Solve x = log2 8 for x. 2x = 8 x = 3 b. logx 25 = 2 x2 = 25 x = 5

Practice c. Solve log2 x = 4 for x. 24 = x x = 16 6.3 Logarithmic Functions c. Solve log2 x = 4 for x. 24 = x x = 16

Example 3 6.3 Logarithmic Functions a. Solve 10x = 14.5 for x. Round your answer to the nearest tenth. log1014.5 = x x = 1.161

Rules and Properties One-to-One Property of Exponential Functions 6.3 Logarithmic Functions One-to-One Property of Exponential Functions If bx = by, then x = y.

Example 4 Find the value of the variable in each equation: 6.3 Logarithmic Functions Find the value of the variable in each equation: a) log2 1 = r b) log7 D= 3 2r = 1 73 = D 20 = 1 D = 343 r = 0

Practice Find the value of the variable in each equation: 6.3 Logarithmic Functions Find the value of the variable in each equation: 1) log4 64 = v 2) logv 25 = 2 3) 6 = log3 v

Practice Solve each equation for x. Round your answers to the nearest hundredth. 1) 10x = 1.498 2) 10x = 0.0054 Find the value of x in each equation. 3) x = log4 1 4) ½ = log9 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: log5 12  1.5440 log5 10  1.4307 log5 120 =  1.5440 + 1.4307  2.9747

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

Example 2 given: log5 12  1.5440 log5 10  1.4307 12 log5 1.2 = log5  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: log5 12  1.5440 log5 10  1.4307 log5 1254 53 = 125 = 4  3 x = 3 = 12

Practice Write each expression as a single logarithm. 1) log2 14 – log2 7 2) log3 x + log3 4 – log3 2 3) 7 log3 y – 4 log3 x

4 minutes Warm-Up Write each expression as a single logarithm. Then simplify, if possible. 1) log6 6 + log6 30 – log6 5 2) log6 5x + 3(log6 x – log6 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) 7log711 – log3 81

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 log2(2x2 + 8x – 11) = log2(2x + 9) for x. 2x2 + 8x – 11 = 2x + 9 2x2 + 6x – 20 = 0 2(x2 + 3x – 10) = 0 2(x – 2)(x + 5) = 0 x = -5,2 Check: log2(2x2 + 8x – 11) = log2(2x + 9) log2 (–1) = log2 (-1) undefined log2 13 = log2 13 true

Practice Solve for x. 1) log5 (3x2 – 1) = log5 2x 2) logb (x2 – 2) + 2 logb 6 = logb 6x

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) 1 2 3 4 5 6 Population 25 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.