Strange Logarithm Phenomenon

Question

I was looking at the following two equations and was trying to solve them when I noticed a very strange phenomenon. These were the equations I was trying to solve: $y=\log_3 x$ & $y=\log_x 3$. I know that multiplying these equations together would give us $y^2=1$ (based on change of base principle). When I try using a different approach, however, I notice that I receive an incorrect answer. Here is my thought process: $3^y=x$ and $x^y=3$ (based on the two equations). If we substitute $3^y$ in for $x$ in the second equation, we will end up with $(3^y)^y=3$. By exponent laws, this is just $3^2y=3$. This gives us the incorrect answer of $y=1/2$. I am fairly sure the problem is in the interpretation of $(3^y)^y$, but how do I know with exponent to deal with first? What do I do?

Answer 1

You're forgetting the rules. $(a^y)^z=a^{y\times z}$, whereas $(a^y)(a^z)=a^{y+z}$.

Answer 2

Your mistake was in $$(3^y)^y=3\implies 3^2y=3$$ instead of $$(3^y)^y=3\implies 3^{y^2}=3$$

Your $y^2 =1$ is correct due to change of base formula $$ \log_ b a = \frac {\ln a}{\ln b}$$