MAT 124 Logarithmic Functions What is a Logarithm? In the early 1600’s, a Scottish mathematician named John Napier introduced a new tool for arithmetical computations: logarithms. This was a revolutionary time-saving device for astronomers at the time. Lengthy calculations that would normally require years could now be done in a few months. This worked because logarithms reduce products and quotients of large numbers into sums and difference of smaller ones. So what is a logarithm? Well, it’s simply the name given to certain exponents. For example, 11 since 23 0.125 , we write log 0.125 3 . Since 54 625 , we write log 625 4 283 2 5 , and so on. Note that logarithms are specific to a given base a that is always positive and not equal to 1. Also, logarithms can only be applied to positive numbers. Thus, loga x is only defined if x 0 . By convention, we write loge xx ln for the natural base e 2.72 and log10 xx log for the common base 10. These widely-used bases are the ones featured on graphing calculators with the LN and LOG keys. Note that any logarithm with a generic base can be written in terms of either the natural base or the common base logarithms using the change-of-base formula: lnxx log log x (See Properties of Logarithms.) a lnaa log Logarithmic Functions By definition, the logarithmic function yx loga is the inverse function of the exponential function ya x . It is defined implicitly by the equation xa y . As such, its domain is the set of positive real numbers and its range is the set of all real numbers. Moreover, we have the following identities: loga x x ax , for x 0 ; loga ax , for any real number x. The graphs of logarithmic functions are either growths ( a 1) or decays ( 01a ). Graph showing logarithmic decay Graph showing logarithmic growth Exercise Find the domain of f( x ) 1 ln x 2 . Graph this function, find its inverse and any x- or y-intercepts. Properties of Logarithms For any base 푎, where 푎 > 0, 푎 ≠ 1, any positive real numbers 푀, 푁, and any real number 푟, the following are always true: 1 푙표푔 1 = 0, 푙표푔 푎 = 1, 푙표푔 ( ) = −1 푎 푎 푎 푎 푙표푔푎(푀푁) = 푙표푔푎푀 + 푙표푔푎푁 *** This is NOT a formula for 푙표푔푎(푀 + 푁) *** 푀 푙표푔 ( ) = 푙표푔 푀 − 푙표푔 푁 푎 푁 푎 푎 푟 푙표푔푎푀 = 푟푙표푔푎푀 푙표푔푏푀 푙표푔푎푀 = 푙표푔푏푎 [This is the change-of-base formula, valid for any other base 푏 such that 푏 > 0, 푏 ≠ 1. Note that in practice this formula is used with 푏 = 푒, the natural base, or 푏 = 10, the common base, since logarithms with those two bases can be computed on a calculator.] These properties are used extensively in exponential & logarithmic models to solve equations. Exercises 2x a) Solve the equations 2푙표푔7푥 = 푙표푔79 and e 5 . b) Show that 푙표푔푎(√푥 + √푥 − 1) + 푙표푔푎(√푥 − √푥 − 1) = 0 for 푥 > 1. c) Approximate the number 푑 that satisfies the equation 3푑 = 4 to the nearest hundredth (this is the fractal dimension of the famous Koch curve). d) Solve the equation 푙표푔5(푥 + 6) = 1 − 푙표푔5(푥 + 2) e) Solve the equation 53xx2 3 2 . f) Solve the equation 4푥 − 2푥 − 12 = 0. [Challenging] Answers 22 a) 2log7x log 7 9 log 7 x log 7 9 x 9 x 3 . Note here that x 3 is NOT a solution since it’s not a positive number. ln5 e22xx5 ln e ln5 2 x ln5 x 0.805 2 b) For 푥 > 1, 푙표푔푎(√푥 + √푥 − 1) + 푙표푔푎(√푥 − √푥 − 1) = 푙표푔푎[(√푥 + √푥 − 1)(√푥 − √푥 − 1)] = 푙표푔푎(푥 − (푥 − 1)) = 푙표푔푎1 = 0 ln 4 c) 3푑 = 4 ⇒ 푙푛(3푑) = 푙푛4 ⇒ 푑 ∙ 푙푛3 = 푙푛4 ⇒ d 1.26 ln3 [Note that this solution can be obtained by substituting the natural base logarithms with common base logarithms.] d) log5x 6 1 log 5 x 2 log 5 x 6 log 5 x 2 1 log5 xx 6 2 1 2 log55xx 8 12 log 5 xx2 8 12 5 xx2 8 7 0 xx 7 1 0 x 1 , since -7 is rejected as an invalid solution because of the domain of log5 x 6 . e) 5x2 3 3 x 2 ln5 x 2 ln3 3 x 2 xx 2ln5 3 2ln3 x ln5 2ln5 3 x ln3 2ln3 x ln5 3ln3 2ln3 2ln5 2 ln3 ln5 x ( 3.212 ) ln5 3ln3 [Check this! Note that this solution can be obtained by substituting the natural base logarithms with the common base logarithms.] f) If we let 2x , then the equation becomes 2 12 0 since x 2 4xx 222 2 . As a result, we have 4 3 0 3,4 . Since is always positive, this implies that 4 2x 4 22 x 2 . .