# Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.2 Derivatives And Integrals Involving Logarithmic Functions - Exercises Set 6.2 - Page 426: 42

$$y = \frac{x}{{10\ln 10}} - \frac{1}{{\ln 10}} + 1$$

#### Work Step by Step

\eqalign{ & {\text{Let }}f\left( x \right) = \log x{\text{ and the point }}{x_0} = 10 \cr & {\text{then }}f\left( {{x_0}} \right) = \log 10 = 1 \cr & {\text{we have the point}}\left( {10,1} \right) \cr & {\text{Find the derivative of }}f\left( x \right) \cr & f'\left( x \right) = \left( {\log x} \right)' \cr & f'\left( x \right) = \frac{1}{{x\ln 10}} \cr & {\text{evaluate }}f'\left( x \right){\text{ at the point }}{x_0}{\text{ to find the slope}} \cr & f'\left( {10} \right) = \frac{1}{{10\ln 10}} \cr & {\text{then }}m = \frac{1}{{10\ln 10}} \cr & \cr & {\text{using the equation of the point - slope form }}y - {y_1} = m\left( {x - {x_1}} \right),{\text{ point }}\left( {10,1} \right) \cr & y - 1 = \frac{1}{{10\ln 10}}\left( {x - 10} \right) \cr & {\text{Simplify}} \cr & y - 1 = \frac{x}{{10\ln 10}} - \frac{1}{{\ln 10}} \cr & y = \frac{x}{{10\ln 10}} - \frac{1}{{\ln 10}} + 1 \cr}

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