Prove that two functions grow at equal rate

Question

Prove that functions $g(x)=\ln(\ln(x))$ and $h(x)=\ln(\lg(x))$ grow at equal rate for every base and value of x.

I'm actually very confused about what 'for every base' actually means. I'm assuming that I'm supposed to keep the outside function as the same, but the inside can be a logarithm with any base?

I can solve this by differentiating both functions, but then again I'm not sure if that's the way to go since it probably doesn't explain the base part. Sorry, if I didn't explain this very well.

Answer 1

Hint: Presumably your $\lg(x)$ is $\log_a(x)$. I believe you are to assume $a,x \gt 1$. Now use the laws of logarithms to express $\log_a(x)$ in terms of $\ln(x)$, then pull out the correction term.

Answer 2

Another hint, assuming that your lg$(x)$ is $\log_{a}(x)$. Note that $\log_{a}(x)=\ln(x)/\ln(a)$...