Linear–log plot

On a linear–log plot, pick some fixed point (x₀, F₀), where F₀ is shorthand for F(x₀), somewhere on the straight line in the above graph, and further some other arbitrary point (x₁, F₁) on the same graph. The slope formula of the plot is:

${\displaystyle m={\frac {F_{1}-F_{0}}{\log _{n}(x_{1}/x_{0})}}}$

which leads to

${\displaystyle F_{1}-F_{0}=m\log _{n}(x_{1}/x_{0})}$

or

${\displaystyle F_{1}=m\log _{n}(x_{1}/x_{0})+F_{0}=m\log _{n}(x_{1})-m\log _{n}(x_{0})+F_{0}}$

which means that

$${\displaystyle F(x)=m\log _{n}(x)+\mathrm {constant} }$$

In other words, F is proportional to the logarithm of x times the slope of the straight line of its lin–log graph, plus a constant. Specifically, a straight line on a lin–log plot containing points (F₀, x₀) and (F₁, x₁) will have the function:

${\displaystyle F(x)=(F_{1}-F_{0}){\left[{\frac {\log _{n}(x/x_{0})}{\log _{n}(x_{1}/x_{0})}}\right]}+F_{0}=(F_{1}-F_{0})\log _{\frac {x_{1}}{x_{0}}}{\left({\frac {x}{x_{0}}}\right)}+F_{0}}$

log–linear plot

On a log–linear plot (logarithmic scale on the y-axis), pick some fixed point (x₀, F₀), where F₀ is shorthand for F(x₀), somewhere on the straight line in the above graph, and further some other arbitrary point (x₁, F₁) on the same graph. The slope formula of the plot is:

${\displaystyle m={\frac {\log _{n}(F_{1}/F_{0})}{x_{1}-x_{0}}}}$

which leads to

${\displaystyle \log _{n}(F_{1}/F_{0})=m(x_{1}-x_{0})}$

Notice that n^log_n(F₁) = F₁. Therefore, the logs can be inverted to find:

${\displaystyle {\frac {F_{1}}{F_{0}}}=n^{m(x_{1}-x_{0})}}$

or

${\displaystyle F_{1}=F_{0}n^{m(x_{1}-x_{0})}}$

This can be generalized for any point, instead of just F₁:

${\displaystyle F(x)={F_{0}}n^{\left({\frac {x-x_{0}}{x_{1}-x_{0}}}\right)\log _{n}(F_{1}/F_{0})}}$