Formulas for calculating points along a straight line if axes are logarithmic

Reading from graphs manually with logarithmic scales is vey error prone.

If you have a graph with a straight line on it where both the x and y axes are in a logarithmic scale the formula of the straight line is found by.

\[\begin{split} \begin{aligned} &X=log(x);Y=log(y);B=log(b)\\ &Y=mX+B\\ &m=\frac{Y_1-Y_0}{X_1-X_0}\\ &m=\frac{log\left(\frac{y_1}{y_0}\right)}{log\left(\frac{x_1}{x_0}\right)}\\ &B=Y-mX\\ &B=log(y_0)-mlog(x_0)\\ &log(x)=\frac{Y-B}{m}\\ &x=10^{\frac{log(y)-B}{m}}\\ &y=10^{mlog(x)+B} \end{aligned} \end{split}\]

Where \((x_0,y_0)\) and \((x_1,y_1)\) are two coordinates along the straight line on the graph.

A similar approach can be taken to derive formulas for a graph where only one of the axes are logarithmic.