logarithms - Power-function relationship becomes linear in log-log space - what does this say about the inverse of the log function?

(apologies that I cannot get the maths formatting to work well!)

If y is a power function of x, i.e.

y=a*(x^b)

this can be written as

lg(y) = lg(a)+b*lg(x)

in other words, that y is a linear function of x in a log-log space.

But this would suggest that the logarithm function is the inverse of the power function, which it isn't! -- the root function is, and the logarithm is the inverse of the exponential function!

Apologies for the banal question, but what am I getting wrong here?

Answer

I don't think the question is silly.

One way of looking at it is: for $y=f(x)$ what functions $g(x),h(x)$ can be applied to $f$ and $y$ such that $h(y)=g(f(x))$ is linear? If $g$ is a logarithmic function and $h$ is linear that means a line on a lin-log graph. If both are logarithmic that means log-log graph.

The inverse of a function means $y=f^{-1}(f(y))$. In other words if $f$ is exponential then $g$ is logarithmic and $h$ is linear. If $f$ is a power function then using it's inverse as $g$ means $h$ would be linear. However I've never heard of a "lin-root" graph, and don't know if such a thing is ever used. However if $g,h$ are both logarithmic (as you pointed out) then you have a line on a log-log graph.

I don't think your intuition was too far off, it was just the y-axis being logarithmic instead of linear that caused the confusion.

Here is a link to Desmos that illustrates, log-lin, lin-log, and log-log graphs. The first entry on the list on the left is the formula to be graphed. You can edit to make it a power function instead of the default exponential function.