Solution to Exercise 6.3-6 © 1997 by Karl Hahn

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The problem is to find the derivative of

   f(x)  =  log10(x)

To solve it you simply use the formula we discussed earlier for converting a log from one base to another. We know how to find the derivative of the natural log, and natural log is log to the base e. So you have to convert  log₁₀  to  log_e. 

Recall that the formula for changing log bases from base a to base b is

logb(x) =

loga(x) loga(b)

So substituting e for a and 10 for b, this formula becomes

log10(x) =

loge(x) = loge(10)

ln(x) ln(10)

And so you finding the derivative of

             ln(x)
   f(x)  =        
            ln(10)

is equivalent to the original problem. And  ln(10)  is just a constant. You know that the derivative of  ln(x)  is  1/x.  So

                1
   f'(x)  =          
             x ln(10)

And that's the answer.

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