We can take a non-linear equation and plot in the form of the equation of straight line,
$$\color{limegreen}y\color{black}=\color{slateblue}m\color{red}x\color{black}+c$$
We will have to compromise in some situations by not plotting the variables $x$ and $y$.
Quadratic equations
Consider the following equation: $$\color{limegreen}y\color{black}=\color{slateblue}a\color{red}x^2\color{black}+b$$ We will have to plot $y$ on the y-axis and $x^2$ on the x-axis: $$\color{limegreen}y\color{black}=\color{slateblue}a\color{red}x^2\color{black}+b$$ $$\color{limegreen}y\color{black}=\color{slateblue}m\color{red}x\color{black}+c$$
Exponential equations
When we have to make these linear, we take the log of both sides: $$ \begin{equation}\begin{aligned} y&=Ae^{2x}\\ \ln{y}&=\ln{Ae^{2x}}\\ \ln{y}&=\ln{A}+\ln{e^{2x}}\\ \ln{y}&=\ln{A}+2x\ln{e}\\ \color{limegreen}\ln{y}\color{black}&=\color{slateblue}2\color{red}x\color{black}+\ln{A}\\ \end{aligned}\end{equation} $$
We will have to plot the natural log of $y$ on the y-axis and $x$ on the x-axis $$ \begin{equation}\begin{aligned} \color{limegreen}\ln{y}\color{black}&=\color{slateblue}2\color{red}x\color{black}+\ln{A}\\ \color{limegreen}y\color{black}&=\color{slateblue}m\color{red}x\color{black}+c\\ \end{aligned}\end{equation} $$
Logarithmic equations
For these, we plot the log of y versus the log of x: $$ \begin{equation}\begin{aligned} \log{y}&=\log{x}+\log{k}\\ \color{limegreen}\log{y}\color{black}&=\color{slateblue}1\times\color{red}\log{x}\color{black}+\log{k}\\ \color{limegreen}y\color{black}&=\color{slateblue}m\color{red}x\color{black}+c\\ \end{aligned}\end{equation} $$