What does this log notation mean?

Question

Can someone please explain what $^2\log x$ means? Is it the same as saying $\log x^2$ or is it something completely different? Here is an image of it as an example:

Answer 1

If $\log_2 \dfrac{x - y}{3} = 0$, then $\dfrac{x - y}{3} = 1 \Rightarrow x - y = 3$.

Now, $\log_4 y = \dfrac{\log_2 y}{\log_2 4} = \dfrac{1}{2}\log_2{y}$.

Then $\log_2 x + 2 \log_4 y = \log_2 x + \log_2 y = \log_2 xy$.

Thus, $\log_2 x + 2 \log_4 y = 2 \Rightarrow \log_2 xy = 2 \Rightarrow xy = 2^2 = 4$.

Now we have, $xy = 4$ and $x - y = 3$, and $x = 4, y = 1$ is an obvious solution.

Therefore, $\boxed{x + y = 5}$.

Note: To solve it in a more general manner-

$x - y = 3 \Rightarrow\\ y = x - 3 \Rightarrow\\ x(x - 3) = 4 \Rightarrow\\ x^2 - 3x - 4 = 0 \Rightarrow\\ (x - 4)(x + 1) = 0 \Rightarrow\\ x = 4, -1 $

We reject $x = -1$ as $\log (-1)$ is not a real number.