How to solve the following equation? $\log_3\big(\log_x(\log_416)\big)=-1$.

Question

$$\log_3\big(\log_x(\log_416)\big)=-1.$$

I am trying to solve this equation for $x$. This is what I have so far:

$$\log_3(\log_x 2)= -1.$$

Okay, now I have this:

log2 = (1/3)logx

How do I isolate x from here?

Answer 1

Adriano's method is another method, I will continue where you stopped:

$$\log_3(\log_x 2)= -1 \Leftrightarrow \frac13=\log_x(2)$$ by the definition of the logarithm.

Now $x^{1/3}=2 \Leftrightarrow x=2^3=8$, again by the definition of the logarithm.

Answer 2

Hint: There is no multiplication going on anywhere in this equation (just nested logs). Try thinking of the stuff inside the brackets as a single unit and convert to exponential form, working your way from the outside to the inside. If it helps, use substitutions. For example, let $u = \log_x(\log_4 16))$, so that we have: $$ \log_3 u = -1 $$ Converting to exponential form, we have: $$ u = 3^{-1} \iff \log_x(\log_4 16) = \frac{1}{3} $$ Now repeat.