Why we equalize the logarithms those with same bases? This is not an exam's question , I already have the answer.
I want to know why when we have the logarithm of something with the same base, we can equalize the things together.
Like here:
$\log_{10} x=\log_{10}y^2\quad$
So  $\quad x=y^2$
Why is this?
 A: This is because the logarithm function with a specific base is injective. For an injective function $f$, it holds that if $f(x) = f(y)$, then $x=y$.
You can prove this in many ways, for example that it is the inverse function of the exponential function.
A: Do you know what the logarithm is?
$$
\log_{10}(x) = a \Leftrightarrow 10^x=a
$$
In other words, $\log_{10}(x)$ asks the question: What do I need to put into the exponent to get $10^{\log_{10}(x)}=x$
Thus, if you now look at
$$\log_{10}(x)=\log_{10}(y^2)$$
you need to ask these questions again. But, because $10^x$ or any other exponential function with a positive base is injective (that means, every y-value is uniquely in correspondence with a fitting x-value), these two questions get answered by the same number and thus
$$x=y^2$$ holds true.
A: Assuming $x, y>0;$
$\log_{10} x=\log_{10}y^2;$
$\log_{10} x-\log_{10} y^2=0;$
$\log_{10} (x/y^2)=0;$
$\Rightarrow x/y^2=1.$
A: The implication you mention is because $x=a^{\log_ax}$, therefore $\log_ay=\log_az$ implies $y=z$.
