change of base formula problem

Question

(a) State the “Change of Base Formula” for logarithms.
(b) If you know that d/dx (ln x) = 1/x, explain (IN WORDS) how you can use the Change of Base Formula to calculate d/dx (log_bx).
(c) Using your explanation in (b), find d/dx (log₄x). Show all your steps.

Answer 1

log_bx = (log_ax)/(log_ab)  b is the given base and a is the base to which we want to convert

 

To find the derivative of log_bx, convert to base e (natural logs):

 

log_bx = lnx/lnb  

 

lnb is a constant, so 1/lnb can be factored out as a constant multiple when taking the derivative of the right hand side

 

We then have:  (log_bx)' = (1/lnb)(lnx)'

 

                                       = (1/lnb)(1/x)

 

                                       = 1/(xlnb)

 

For example, (log₄x)' = 1/(xln4)