= Semi-log plot
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= Semilog plot
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A <semi-log plot> uses a <logarithmic scale> on one axis and a <linear scale> on the other. Which axis is logarithmic matters. Plotting $x$ against $\log y$ makes positive <exponential functions> appear as <straight lines>; plotting $\log x$ against $y$ instead makes an affine function of a <logarithm> appear straight. The plotting operations change positions while retaining the original tick values.<Matplotlib 3.10.7 documentation/Log scale>[Matplotlib 3.10.7 documentation; Matplotlib 3.10.7, Log scale: Convenience functions semilogx, semilogy, and loglog; distinction between horizontal and vertical logarithmic axes]{ref}
Take the original dimensionless <mathematical model> $y(t)=3e^{0.4t}$ for $0\leq t\leq8$. It has <exponential growth>, with $y(0)=3$, $y(4)\approx14.8591$, and $y(8)\approx73.5976$. Taking a base-ten <logarithm> gives
$$
\log_{10}y=\log_{10}3+\frac{0.4}{\ln10}t.
$$
The <slope> in these transformed positions is $0.4/\ln10\approx0.173718$, although the original relative growth rate $y'/y$ is $0.4$. Equal steps of two in $t$ multiply $y$ by $e^{0.8}\approx2.22554$. The corresponding ordinary gains increase, so the left plot bends while the right plot is straight.
\Image[/logarithmic-scale-exponential-growth-axes.png]
{title=The same <exponential growth> model bends on a <linear scale> and is straight on a <semi-log plot> with a logarithmic vertical axis}
{description=Original exact <mathematical model> $y(t)=3e^{0.4t}$, not measurements. Both panels show identical values at the same $t$. Right-hand tick labels retain the original $y$ values, with equal gaps for each doubling. The stated <slope> is for $(t,\log_{10}y)$, not the original <derivative> $y'$.}
One practical use is checking a proposed <exponential distribution> for failure times. For rate $\lambda>0$ and $t\geq0$, its <cumulative distribution function> is $F(t)=1-e^{-\lambda t}$, so the reciprocal <survival function> $y=1/[1-F(t)]$ satisfies $\log_{10}y=\lambda t/\ln10$. Estimated <probabilities> can therefore be placed on a logarithmic vertical axis and compared with a <straight line>. Agreement supports compatibility with the chosen <statistical model>; a roughly straight finite sample does not establish a unique physical mechanism.<NIST and SEMATECH e-Handbook of Statistical Methods/8.2.2.1. Probability plotting>[NIST and SEMATECH e-Handbook of Statistical Methods, §8.2.2.1 Probability plotting: Exponential Model, logarithmic vertical-axis transformation and interpretation of approximate straightness]{ref}
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