# What happens in a log log graph at the origin where $x = 0$?

I'm using a publicly available textbook to revise some maths and learn a bit of basic astrophysics. The section on logarithms discusses power laws and log log graphs.

It uses the following generalised example of $$y = ax^k$$ can be plotted as $$\log y = \log a + k \log x$$. This results in a straight line graph where the gradient is equal to $$k$$ and the intercept gives the value of $$\log a$$. But how can there be an intercept? When $$x = 0$$ isn't $$k \log x$$ undefined?

At the moment I'm just pretending that it's a trick that we play so that we can use a useful graph to infer the result rather than a precise calculation? i.e. $$\log x$$ is effectively $$0$$ at the origin rather than undefined.

Textbook extract

• If you read the given section closely, the graph is between $\log y$ and $\log x$. The intercept is defined by putting one of the axis as 0. In this case, the "$y$-intercept" is actually the $\log y$ intercept and is found out by putting $\log x$ (the other axis) as zero, not $x =0$. May 11 at 17:35
• That took me a minute and a few reads to flip my perspective round but I see it now. Thank you Adam and Joe for both taking the time to explain it, and to whoever fixed my formatting for me! May 12 at 0:22

As Adam mentions in the comments, you are plotting $$\log y$$ against $$\log x$$, and so the intercepts actually occur when $$\log y=0$$ and when $$\log x = 0$$. For ease of notation, let $$X=\log x$$ and $$Y=\log y$$. The equation $$\log y = k\log x + \log a$$ becomes $$Y = kX + \log a \, ,$$ which you can recognise as the graph of a linear equation. The $$Y-$$intercept (what is called the 'y-intercept' in the testbook) occurs when $$X=0$$, so $$\log x = 0$$. This equation has the solution $$x=1$$, regardless of the base of the logarithm used.