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−1(+)00(−)90(+)y=f(x) =xlog(x+ 1) andy=g(x) =x
464Exponential and Logarithmic FunctionsOur next example revisits the concept of pH as first introduced in the exercises in Section6.1.Example 6.4.3.In order to successfully breed Ippizuti fish the pH of a freshwater tank must beat least 7.8 but can be no more than 8.5.Determine the corresponding range of hydrogen ionconcentration, and check your answer using a calculator.Solution.Recall from Exercise77in Section6.1that pH =−log[H+] where [H+] is the hydrogenion concentration in moles per liter. We require 7.8≤ −log[H+]≤8.5 or−7.8≥log[H+]≥ −8.5.To solve this compound inequality we solve−7.8≥log[H+] and log[H+]≥ −8.5 and take theintersection of the solution sets.3The former inequality yields 0<[H+]≤10−7.8and the latteryields [H+]≥10−8.5.Taking the intersection gives us our final answer 10−8.5≤[H+]≤10−7.8.(Your Chemistry professor may want the answer written as 3.16×10−9≤[H+]≤1.58×10−8.)After carefully adjusting the viewing window on the graphing calculator we see that the graph off(x) =−log(x) lies between the linesy= 7.8 andy= 8.5 on the interval [3.16×10−9,1.58×10−8].The graphs ofy=f(x) =−log(x),y= 7.8andy= 8.5We close this section by finding an inverse of a one-to-one function which involves logarithms.Example 6.4.4.The functionf(x) =log(x)1−log(x)is one-to-one.Find a formula forf−1(x) andcheck your answer graphically using your calculator.Solution.We first writey=f(x) then interchange thexandyand solve fory.y=f(x)y=log(x)1−log(x)x=log(y)1−log(y)Interchangexandy.x(1−log(y))=log(y)x−xlog(y)=log(y)x=xlog(y) + log(y)x=(x+ 1) log(y)xx+ 1=log(y)y=10xx+1Rewrite as an exponential equation.3Refer to page4for a discussion of what this means.
6.4 Logarithmic Equations and Inequalities465We havef−1(x) = 10xx+1. Graphingfandf−1on the same viewing window yieldsy=f(x) =log(x)1−log(x)andy=g(x) = 10xx+1