Finding a specific point on a logarithmic scale

Question

I've been racking my head around this issue for a while now, watching videos and searching on Google, but to no avail.

What I am trying to do is to find an equation that enables me to note with a pencil on a drawn-to-scale logarithmic scale, where a certain number is. For example, in the picture below, I'd like to find out which point on the scale will I be able to note the number $20$.

Hand-drawn log scale -

Thanks!

Answer 1

The tick values are given by

$$3^i$$ where $i$ is the tick index, starting at $i=1$ on the left.

Now the equation

$$3^x=20$$

has the solution

$$x=\log_320\approx2.73$$ so that the number $20$ lies at $73\%$ of the second interval (about third quarter).

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As you can check, $$3^{2\frac34}=9\sqrt{\sqrt{3^3}}\approx20.5$$

Answer 2

Hint:

How do you know that the number $3$ is one unit to the right of the origin? Answer: $\log_33$

How do you know that the number $9$ is two units to the right of the origin? Answer: $\log_39$

So, how do you know where the number $20$ is on the scale?