Confusion about exponents in logarithmic equation

Question

In this question: $$5^{(\log_5x)^2}+x^{\log_5x}=1250$$ the term in the LHS of the question is $5^{(\log_5x)^2}$. My perhaps dumb question is, does this mean we're squaring $\log_{5}{x}$ and then raising 5 to that power? Or are we raising 5 to $\log_{5}{x}$ and then squaring that whole thing? From the solution given in both the answers, clearly it's the second one. But why? If it was the first option, would it have been notated differently? I would appreciate some clarity on this. :)

Answer 1

does this mean we're squaring $\log_{5}{x}$ and then raising 5 to that power?

Yes.

From the solution given in both the answers, clearly it's the second one.

On the contrary, it's very clearly the first option. For example, in one of the answers,

$$ 5^{(\log_5x)^2} = (5^{(\log_5x)})^{\log_5x}, $$

but

$$ (5^{(\log_5x)})^{\log_5x} = 5^{((\log_5x)\cdot(\log_5x))} = 5^{((\log_5x)^2)} $$

using the rule that $(n^a)^b = n^{(a\cdot b)}. $

Answer 2

No, in both answers it’s the first option: $$ 5^{(\log_5 x)^2} = 5^{(\log_5 x) (\log_5 x)} = (5^{\log_5 x})^{\log_5 x} $$

This convention makes sense because we can write $(a^b)^c$ as $a^{bc}$ whereas there no similar simplification for $a^{(b^c)}$.