Does $\log_y x=1$ create the line $y=x$?

Question

Given the equation $\log_y x = 1$ can I say that it creates the line of $y=x$?

I know that the base of a log cannot be negative as any even root of the number would be undefined but that would only apply to an equation such as $\log_a x = y$. Furthermore using methods such as the change of base gives me $1 = \frac{log(x)}{log(y)}$ and that would be undefined. But I don't understand how just converting $\log_y x = 1$ to $y^1 = x$ wouldn't work.

So would it being previously in logarithmic form bring some sort of restriction I'm missing out on?

Thank you

Answer 1

$1=(\log x)/(\log y)$ isn't undefined. Just note that, in any base, the logarithm is a real number only if its argument is positive. We can manipulate $1=(\log x)/(\log y)$ just fine and get $\log x=\log y$, hence $x=y$, but we must put the restrictions $x>0$ and $y>0$. Technically, this does not create a line, but a ray with a hole at the origin.