I'm working on the problem below, and I would be extremely thankful if anyone could provide pointers or corrections to what I've done so far.
Here is the question:
Using a base $10$ log-linear plot, if $x = 10$ and $x = 1000$ are $1$ cm apart, what is the distance between $x = 1000$ and $x = 1500$? (In other words, where—when drawn on a log base $10$ scale—do $x = 10, 1000,$ and $ 1500 $ correspond to the analogous linear number line?)
I was thinking of making a ratio between the two and solving for $x$, such as $\frac{0.01}1=\frac{0.66}x)$, but I don't think I've done it correctly. Again, any help would be much appreciated! Thank you for taking the time to read my question.
The distance should be linearly proportional to the difference in the base-10 logarithms of the two values of $x$.
You are given that as the base-10 logarithm goes from $(1)$ to $(3)$ that the distance is $(1)$ cm.
So, given that $\log_{10}(1500) \approx 3.176$, you have that the distance between $(1500)$ and $(1000),$ measured in cm, should approximately be
$$\frac{1}{3 - 1} \times \left[3.176 - 3.000\right] = \frac{0.176}{2}.$$