The logarithmic function to the base 10 is called the common logarithmic function . 3 Example 1: (a) Solve log8 = 1 x for x. (b) Solve 5 0.04 for = x. x (a) = 2 (b) = 2 x x Differentiation and Integration Use the differentiation rules for bases other than To differentiate exponential and logarithmic functions to e. bases, you have three options: What you should How to differentiate and Use the definitions of and loga and differentiate the ax x integrate exponential rules for the natural exponential and logarithmic functions. functions that have bases other than Use logarithmic differentiation, or e Let be a positive real number (a 1) a and let be a u differentiable function of Complete the following formulas for x. the derivatives for bases other than e. d = (ln . a) ax dx d (ln [du/dx] . = a) au dx [log = 1 / [(ln ] . d x] a) x dx a [log 1 / [(ln ] [du/dx] . = d u] a) u dx a Occasionally, an integrand involves an exponential function to a base other than When this occurs, there are two options: e. (1) convert to base e the formula = and ax e(ln a)x then integrate or (2) integrate directly using the = integration formula dx (1/ln + . a) ax C
The logarithmic function to the base 10 is called the common logarithmic function . Example 1: (a) Solve log8 x = 1 3 for x. (b) Solve 5 x = 0.04 for x. (a) x = 2 (b) x = 2 Differentiation and Integration To differentiate exponential and logarithmic functions to bases, you have three options: Use the definitions of ax and loga x and differentiate the rules for the natural exponential and logarithmic functions. Use logarithmic differentiation, or Use the differentiation rules for bases other than e. What you should How to differentiate and integrate exponential functions that have bases other than e Let a be a positive real number (a 1) and let u be a differentiable function of x. Complete the following formulas for the derivatives for bases other than e. d = (ln a) ax . dx d = (ln a) au [du/dx] . dx d [log x] = 1 / [(ln a) x ] . dx a d [log u] = 1 / [(ln a) u ] [du/dx] . dx a Occasionally, an integrand involves an exponential function to a base other than e. When this occurs, there are two options: (1) convert to base e the formula ax = e(ln a)x and then integrate or (2) integrate directly using the integration formula dx = (1/ln a) ax + C .
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