Why we equalize the logarithms those with same bases?

Question

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?

Answer 1

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.

Answer 2

The implication you mention is because $x=a^{\log_ax}$, therefore $\log_ay=\log_az$ implies $y=z$.

Answer 3

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.

Answer 4

Assuming $x, y>0;$

$\log_{10} x=\log_{10}y^2;$

$\log_{10} x-\log_{10} y^2=0;$

$\log_{10} (x/y^2)=0;$

$\Rightarrow x/y^2=1.$