More Derivatives of Logarithms - At A Glance

Find the derivative of  f(x) = log₁₀ x.

We can rewrite this logarithm as

Since (ln 10) is a constant, the function f is a constant multiple of ln x:

We know how to find the derivative of this function:

Find the derivative of f(x) = log₄ x².

First, simplify the original function. We need to bring the exponent down in front, split up the logarithm, and combine all the constants into one:

Or, we could instead split up the logarithm first and then bring down the exponent (the order of these two operations doesn't matter):

Now we have a nicely rewritten function:

We've done an awful lot of work and haven't even started taking the derivative yet, but now, taking the derivative will be a piece of cake. The function f is a constant multiplied by ln x, therefore

We could make this even tidier by recognizing that
ln 4 = ln (2²) =  2 ln 2,
therefore