When is x +/- and when is it just positive when solving logarithms

Question

I have two similar questions that I've solved to the best of my ability.

The first question is

log2(9)=2

9 = x^2

√9=x

3=x

The second one is log(x^2)-15 = 1

x^2-15=10

x^2 = 25

√25=x

+/-5 = x

My question is, why is one of them just a positive for the x, and why is the other one plus or minus for x?

I checked the answer in both my textbook and wolfram alpha, and the answer is consistent with what I've written here.

Answer 1

I guess that the first one is $\log_x(9)=2$, so $x^2=9$ and $x=3$, because the base of a logarithm must be positive.

If the unadorned $\log$ denotes decimal logarithm, the second equation is $$ \log(x^2)=16 $$ or $$ 2\log|x|=16 $$ hence $|x|=10^8$. If it's, instead, $\log(x^2-15)=1$, then it becomes $$ x^2-15=10 $$ so $x=5$ or $x=-5$. Both solutions are good, because in both cases $x^2-15>0$.

The rules

1. The base of a logarithm must be a positive number different from $1$.
2. The argument of a logarithm must be a positive number.