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Sullivan , 8th ed. MAC1140/MAC1147 6-4: Logarithmic Functions The inverse function of y = f(x) = ax is x = ay, which is equivalent to the logarithmic function y = log a x. This equivalence is used to convert between logarithmic and exponential functions. If the base is the number e, then x = e y if and only if y = ln x. Write the equation in logarithmic form: Write in exponential form: 2 2.2 1. 16 = 4 2. e = M 3. logb 4 = 2 4. ln x = 4 Evaluate; if au = av, then u = v: 5. log5 3 25 6. ln e3 7. Find a so that the graph of f(x) = logax. contains the point (32, 8); simplify. y = loga x or x = ay a>1 Domain Range x-intercept [0, ) (-, ) (1, 0) 0<a<1 [0, ) (-, ) (1, 0) Vertical asymptote y-axis as y - y-axis as y Characteristic Increasing, one-to-one Decreasing one-to-one Passes through (1, 0) (a, 1) (1, 0) (a, 1) Graph the given logarithmic function and its inverse; state the domain, range, and any vertical asymptote: 8. f(x) = 3 – ln (x + 2) x e3-y - 2 0 1 2 3 4 e(3 - x) 9. f(x) = 2 + log1/2 x -2 =y x= (1/2)y-2 -1 0 1 2 3 (1/2)x-2 =y Sullivan , 8th ed. MAC1140/MAC1147 6.5: Properties of Logarithms Properties: (1) loga 1 0 and loga a 1 ; (2) a loga M M and log a a r r ; M log a M log a N ; (5) log a M r r log a M ; (3) loga MN loga M loga N ; (4) log a N (6) log a 1 log a N . If M, N, and a are positive real numbers and a 0 and b0, then (7) if M = N N, loga M loga N , and (8) if loga M loga N , M = N. Change of base formulas: (9) log a M log b M and (10) log a M ln M . log b a ln a Use the properties of logarithms to find the exact value of each expression. Do not use a calculator: 1. ln e log7 15log7 3 2. log 6 9 log 6 4 2 3. 7 Use properties of logarithms to rewrite the logarithm in terms of p, q r, or s: 4. If ln 4 = p, 5. If ln 10 = q and ln 18 = r, 6. If ln5 = r and ln 225 = s, ln 256 = ln1.8 = ln 3 45 = Write as the sum or difference of logarithms: 2 5 7. logb 3 x y6 2 z 8. x 4 2 3 ln 2 x 1 Express as a single logarithm: 9. logb 2x 3 logb x logb y 10. 35 log 4 7 2x 2 log 4 5x 3 log 4 3 Sullivan , 8th ed. MAC1140/MAC1147 Evaluate; round your answer to nearest hundredth. 11. log5 18 13. log5 343 . log725 12. log 2 14. log24 . log4 6 . log6 8 Express y as a function of x. C is a positive number. 15. ln y = 15x + lnC 1 3 1 5 16. 5ln(y) = ln( x 5) ln x 7 lnC Sullivan , 8th ed. MAC1140/MAC1147 Try These (6.4-6.5) Use the properties of logarithms to find the exact value of each expression. Do not use a calculator: 1. Graph the function: y = -log(x – 3) + 2 Domain: {xx > 3} Range: (-, ) Asymptotes: x = 3 2. a. log816 – log82 log88 = 1 b. e ln 29 x x = 29 Write as the sum or difference of logarithms: x3 x 1 1 3. log x 2 2 3log x 2 log( x 1) 2log( x 2) Express as a single logarithm: 4. log x 3 x 1 x2 2 x 3 x2 7 x 6 x2 l og log 2 x 4 x2 x 2 x 2 x 6 x 1 x 3 x 1 x 3 x 1 1 log log x 2 x 6 x 1 x 2 x 6 x 1 Express y as a function of x. The constant C is a positive number. 5. log(y + 8) = 3x + log C log (y + 8) – log C = 3x y8 log 3x C y8 103 x C y + 8 = 103xC y = 103xC - 8