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Inverse of An Exponential Function Since an exponential function, f(x) = bx, is an 1-1 function, it has an inverse. 1 To find the inverse, we need to solve x b f ( x) . The solution is defined to be f -1(x) = logbx. • This is a logarithmic function with base b. • b has the same restrictions as the exponential function, namely b > 0 and b ≠ 1. • f-1(x) is the power b must be raised to in order to get x. Examples log232 log2 1 4 log55 log 1 log41 2 1 8 log 1 4 2 A Logarithmic Function Form: f(x) = logbx where b > 0 and b ≠ 1. Domain: x > 0. Range: All real numbers. Vertical Asymptote: x = 0. Intercepts: (1, 0) Increasing for x > 0 provided b > 1. Decreasing for x > 0 provided 0 < b < 1. No extreme values. Graphs: Special Logarithms The common logarithm, logx, is the logarithm with base 10. logx = log10x The natural logarithm, lnx, is the logarithm with base e. lnx = logex Examples log100 log0.1 log67 ln(e3) ln9 ln0.245 log214 Transformations All of the transformations apply. Examples 1. f ( x) 3log 2 ( x 5) 2 2. f ( x) 2ln( 3x) 1 Properties of Logarithms 1. logb1 = 0 Rationale: b0 = 1. 2. logbb = 1. Rationale: b1 = b 3. logbbx = x for all real numbers. Why is it true for all real numbers? Rationale: Definition of inverses. logb x 4. b x for x > 0. Why is it true only for x > 0? Rationale: Definition of inverses. Implication: Any exponential function can be converted to an exponential function in another base. • Examples a. Since 2 = eln2, 2 x b. Since 3 5log5 3 , 3x eln 2 x 5log5 3 e(ln 2) x x 5. logbx = logby if and only if x = y. Rationale: 1-1 Property of Logarithms. e0.693 x . 5(log5 3) x 50.6826 x . Change of Base Formula log a x log b x . log a b log x and logb x logb x log b Examples log214 log350 ln x are more practical. ln b log567 log20.75 log50.01 log 1 483 2 log 1 0.1 2 Additional Properties of Logarithms Let x and y be positive numbers and n be any number. 1. logb(xy) = logbx + logby Proof: a. Let p = logbx and q = logby. b. By the definition of logarithms, x = bp and y = bq. c. Using the laws of exponents, xy = bpbq = bp + q. d. Finally, logb(xy) = logb(bp + q) = p + q = logbx + logby. 2. logb(x/y) = logbx – logby Proof: Similar to above. 3. logb(xn) = nlogbx Proof: Similar to above. Examples 6 log4x2y3 log5 2ln8 + 5ln(z – 4) 5log3x + 4log3y – 7log3z x2 1 3 log 2 x2 y 4 2z3 ½[log(x + 1) + 3log(x – 1)] + 6logx