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 loge.
Recall that the formula for changing log bases from base a to base b is
loga(x)
logb(x) =
loga(b)
So substituting e for a and 10 for b,
this formula becomes
loge(x) ln(x)
log10(x) = =
loge(10) 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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