The graph of $y= 3^{x+2}$ is reflected in the line $y=x$, the equation of the new graph is? ($y=\log_3(x-2)$ OR $y=\log_3 x-2$)

Question

The graph of $y=3^{x+2}$ is reflected in the line $y=x$, the equation of the new graph is? ($y=\log_3(x-2)$ OR $y=\log_3 x-2$)

Hi, I don't understand what the difference is in these answers? Which is correct? Should the $x-2$ be in brackets or is that improper? The $\log$ is base 3 by the way. Thanks so much!

Answer 1

We have $$y=3^{x+2}$$ to be reflected along $y=x$ so replace $x$ with $y$ and $y$ with $x$ in original question to get $$x=3^{y+2}$$ $$\Rightarrow \log_3 x=y+2$$ $$\Rightarrow y=(\log_3 x) -2$$