Differentiation of Logarithmic Functions The rule for finding the derivative of a logarithmic function is given as: If y log a then dy or y. d a ( ln This rule can be proven by rewriting the logarithmic function in eponential form and then using the eponential derivative rule covered in the last section. y log a Begin with logarithmic function y a ln a a y ln a Convert into eponential form Differentiate both sides of the equation Substitute for the eponential function a y Solve for y by dividing each side by (ln a a ( ln As with the eponential rules, the derivative of a logarithmic function can be simplified if the base of the logarithm is e. y log ln Logarithmic function with base e e Apply the logarithm derivative rule above e ( ln Use properties of logarithms to simplify ( Gerald Manahan SLAC, San Antonio College, 008
Eample : Find the derivative of log6 f log 6 ( f ( ln 6 f Eample : Find the derivative of f ( log( Right now the only derivative rule we have for logarithms is for log. So in order to take the derivative of this function we must first use the properties of logarithms to rewrite the function. First use the product property of ( y log( log( + log( f log log + log y r Net use the eponent property of log( r log( log( + log f log( log + Now we can find the derivative. log( + log f f ( 0+ ln0 ( ln0 Even though we were able to find the derivative of this function, it would be easier to find the derivative if we had a rule that dealt with the situation where a function is equal to the log of another function. In order to do this, we would have to combine the chain rule with the logarithm rule. Gerald Manahan SLAC, San Antonio College, 008
Lets look at an eample similar to the first function that we looked at when proving the logarithm derivative rule. [ ] g f g log b Composite logarithmic function b [ ] f g g Convert to eponential form [ ] [ ] f g ln b b f g( g Differentiate both sides of the equation ( ln b g( f [ g( ] g ( Substitute g( for the eponential function [ g( ] f g ( ( ln b g( Divide each side by (ln b g( Derivative of g( log b If y g( log b then ( g ln b g If y ln g( then g ( g( Now lets go back to eample and find the derivative again but this time using the above rule. Gerald Manahan SLAC, San Antonio College, 008
Eample : Find the derivative of f ( log( Note: Remember if a base is not shown it is understood to be 0. log( f D ( f ( ln0 8 ln0 ln0 ( ln0 As you can see we get the same derivative as before but in an shorter and easier process. Eample : Find the derivative of y ln ( t + 5t In this problem we can simplify the derivative process by first applying the properties of logarithms to move the eponent out in front of the function as a coefficient. ( t t y ln t + 5t ln + 5 Now we can find the derivative y ln t + 5 t D ln ( t + 5t Gerald Manahan SLAC, San Antonio College, 008
Eample (Continued: ( + 5 ( t + t D t t 5 ( t + 5 ( t + t 5 Now lets look at a more comple function that will require the use of several derivative rules. Remember when finding the derivative of a comple function take it step by step. Don t try to do it all at once. Eample 5: Find the derivative of y ( + e ( + ln 5 Since the function is in the form of a fraction we must begin by applying the quotient rule. When you go to find the derivative of the numerator you are will have to use both the product and eponential rules. The derivative of the denominator will require the use of the logarithm rule. First, apply the quotient rule. y ( + e ( + ln 5 ( + D ( + e ( + e D ( + ln 5 ln 5 ln ( 5 + Gerald Manahan SLAC, San Antonio College, 008 5
Eample 5 (Continued: Net, apply the product and eponential rule to the derivative of the numerator. ( + D ( + e ( + e D ( + ln 5 ln 5 ln ( 5 + ( + ( + D e + e D ( + ( + e D ( + ln 5 ln 5 ln ( 5 + ( + ( + e D ( + e ( ( + e D ( + ln 5 ln 5 ln ( 5 + ln 5+ + e + e + e D ln 5+ ln ( 5 + Now, apply the logarithmic rule to find the derivative of the denominator. ( + ( + ( e + e ( + e D ( + ln 5 ln 5 ln ( 5 + ( 5+ ( 5 D ln ( 5+ ( ( e e + + ( + e + ln ( 5 + 5 ln ( 5+ ( ( e e + + ( + e + ln ( 5 + ( 5 The only thing left to do now is to simplify the derivative using the properties of algebra. Gerald Manahan SLAC, San Antonio College, 008 6
Eample 5 (Continued: 5 ln ( 5+ ( ( e e + + ( + e + ln ( 5 + e + + ln 5+ + e ( + ln 5 ( + + ( + ( + ( + ( 5 + e 5 ln 5 5e ( 5+ ln( 5+ ( + + ( + ( + ( + e 5 ln 5 5 ( 5+ ln( 5+ 5 ( 5 Gerald Manahan SLAC, San Antonio College, 008 7