The general log rule, and inverse functions

The general log rule relates the log function to an exponential function

The general log rule that we introduced earlier was

Given the equation $a^x=y$, the associated log is $\log_a{(y)}=x$, and vice versa.

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What this tells us is that

$\log_a{(y)}=x$ and $a^x=y$ are equivalent

$\log_a{(x)}=y$ and $a^y=x$ are equivalent

Remember that inverse functions have their $x$- and $y$-values swapped. This means that when you graph inverse functions on the same set of axes, the graphs are mirror images of one another, just reflected over the line $y=x$.

We can see that $\log_a{(y)}=x$ and $\log_a{(x)}=y$ have their $x$- and $y$-values swapped, and that $a^x=y$ and $a^y=x$ have their $x$- and $y$-values swapped. Which means that

Both $\log_a{(x)}=y$ and $a^y=x$ are inverses of $\log_a{(y)}=x$

Both $\log_a{(x)}=y$ and $a^y=x$ are inverses of $a^x=y$

Both $\log_a{(y)}=x$ and $a^x=y$ are inverses of $\log_a{(x)}=y$

Both $\log_a{(y)}=x$ and $a^x=y$ are inverses of $a^y=x$

For example, the graph of $\log_a{(x)}=y$ (or equivalently $a^y=x$) is

And the graph of $\log_a{(y)}=x$ (or equivalently $a^x=y$) is

And we can see that these are inverses of one another, because they are a reflection of each other over the line $y=x$.

When functions are inverses of one another, we can also express their points in tables. For instance, given the equations $a^x=y$ and $\log_a{(x)}=y$, we can express points that satisfy each of these equations in tables.

If a point set that satisfies $a^x=y$ is

then the point set satisfying its inverse $\log_a{(x)}=y$ is

And if we sketch these points on a graph, we can see again how they are mirror images of one another over the line $y=x$.

How to convert log functions to exponential functions, and vice versa

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