Center of range on logaritmic range

Question

Suppose I have a range between 500 and 20.000. How do I find the centre of this range on a logaritmhic scale?

Thanks in advance

Answer 1

If instead, the upper limit were $200,000$ and we averaged the log's, we'd get

$$\frac{\log_{10} 500+\log_{10} 200000}{2} = \frac{\log_{10} 10^8}{2} =4 \log_{10} 10 =\log_{10} 10000.$$

Which makes sense to me. I can't make sense of $20,000$.

Answer 2

On a logarithmic scale, each value $x$ is plotted at a position proportional to $\log(x)$. Therefore, the average of the positions for $500$ and $20000$ would be $k\log(500)$ and $k\log(20000)$. The average of these positions is $\frac k2\log(10000000)$ which would be the position corresponding to $\bbox[5px,border:2px solid #C0A000]{1000\sqrt{10}\approx3162.28}$.