5.4 "Logarithmic Functions" 32 = 2x 3 = 27x 100 = 5x How do we solve for the variable? y = loga x if and only if x = ay Logarithmic Form: Exponential Form: a y = x log a x = y (ex ) Change to exponential form: ("A logarithm is an exponent") a) log 3 81 = 4 b) log 7 7 = 1 c) log 14 1 = 0 d) log 1/2 32 = 5 (ex) Change to logarithmic form. a) 2 3 = 8 b) 4 0 = 1 c) 12 1 = 12 d) (1/4) 1 = 4 "The logarithm of 8 with base 2 is 3" exponent (try) Change to exponential form. a) log 3 81 = 4 b) log 4 (1/256) = 4 c) log v w = q d) log 6 (2x 1) = 3 e) log 4 P = 5 x f) log a 343 = 3/4 (try) Change to logarithmic form. a) 3 5 = 243 b) 3 4 = 1/81 c) c p = d d) 7 x = 100p e) 3 2x = p/f f) (0.9) t = 1/2 . 1 Solve for t using logarithms with base a (ex) 3a4t = 10 (ex) F = D + Bat Common Logarithm (ex) L = Mat/n P Natural Logarithm log10 x log x loge x ln x (ex) Change to logarithmic form. a) 102 = x b) 10y+3 = z c) e2 = x d) ey+3 = z (try) Change to logarithmic form. a) 104 = 10,000 b) 102 = 0.01 c) 10x = 38z d) e4 = D e) e0.1t = x+2 (try) Change to exponential form. a) log x = 8 b) log x = y2 c) ln x = 1/2 d) ln z = 7+x e) ln(t5) =1.2 (ex) Find the number: a) log 10 100 b) log 2 (1/32) d) log 7 1 e) log 3 (2) g) log 5 0.2 . h) log 1/5 125 c) log 9 3 f) log 4 64 2 Some General Properties: 1. log a 1 = 0 because a0 = 1 (ex) log 7 1 = 0 2. log a a = 1 because a1 = a (ex) log 2 2 = 1 3. log a ax = x because ax = ax (ex) log 10 104 = 4 (ex) log39=log332=2 4. because (look at example) (ex) (ex) (ex) (ex) Find the number. a) 10 log 7 b) log 106 c) log 100,000 d) log 0.001 e) e ln 8 f) ln e g) e 1 + ln 5 Solve the equation. (ex) log 3 (x + 4) = log 3 (1 x ) (ex) log 7 (x 5) = log 7 (6x) (ex) ln x2 = ln (12 x) (ex) log 2 (x 5) = 4 (ex) log 4 x = (ex) log x2 = 4 . 3 (ex) e ln x = 0.2 (ex) e x ln 2 = 0.25 Approximate x to three significant figures. a) log x = 1.8965 b) log x = 4.9680 c) log x = 2.2118 d) ln x = 3.7 e) ln x = 0.95 f) ln x = 5 Using your calculator: 10x = 251.19 10x = 900 ex = 90 Doubling Time. A population is growing continuously at a rate of 4% per year. What's its doubling time? (growth formula: q = q0ert ) Bismuth Decay. The radioactive bismuth isotope 210Bi disintegrates according to Q = k(2)t/5 , where k is a constant and t is the time in days. Express t in terms of Q and k. Population Growth. The population N(t) (in millions) of India t years after 1985 may be approximated by the formula N(t) = 766e0.0182t . When will the population reach 1.5 billion? Continuously Compounded Interest. If interest is compounded continuously at the rate of 6% per year, approximate the number of years it will take an initial deposit of $6000 to grow to $25,000. . 4

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