= Logarithmic midpoint
{title2=$m=\sqrt{ab}$}
For positive endpoints $a,b$, halfway along a <logarithmic scale> means
$$
\log_{\beta}(m/x_0)=\frac{\log_{\beta}(a/x_0)+\log_{\beta}(b/x_0)}2.
$$
Here $\beta>1$ is the base. Equivalently, using the <natural logarithm>, $2\ln(m/x_0)=\ln(a/x_0)+\ln(b/x_0)$. The <product of numbers> identity for <logarithms> gives $m^2=ab$, so \b[the midpoint is the <geometric mean> $\sqrt{ab}$]. The two endpoints lie at equal multiplicative distances because $m/a=b/m$.
For $a=10$ and $b=100$, this midpoint is $\sqrt{1000}\approx31.6228$, giving a factor of $\sqrt{10}$ on each side. The <arithmetic mean> $55$ is halfway on a <linear scale>, but its two <ratios> are $55/10=5.5$ and $100/55\approx1.81818$. Thus reading halfway between the marks $10$ and $100$ as $55$ on a <logarithmic scale> gives the wrong value.
\Image[/logarithmic-scale-ratios-and-midpoints.png]
{title=Equal <ratios> give equal logarithmic gaps, and the <geometric mean> is halfway between two positive values}
{description=Original rulers calculated from <logarithms>. The lower rulers cover the same endpoints $10$ and $100$: the blue <geometric mean> is centered on the <logarithmic scale>, while the orange <arithmetic mean> is centered on the <linear scale>. Tick labels show original values.}
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