# How do you use the formula (ln f(x))'= f'(x) / f(x) to show that lnx and ln(2x) have the same derivative?

Jun 9, 2016

We know that:

$\left(\ln f \left(x\right)\right) ' = \frac{f ' \left(x\right)}{f} \left(x\right)$

So, for $\ln x$, we see that $f \left(x\right) = x$, which implies that $f ' \left(x\right) = 1$.

Thus:

$\left(\ln x\right) ' = \frac{1}{x}$

For $\ln \left(2 x\right)$, we see that $f \left(x\right) = 2 x$, so $f ' \left(x\right) = 2$.

Thus:

$\left(\ln \left(2 x\right)\right) ' = \frac{2}{2 x} = \frac{1}{x}$