How would you do that without a calculator, so that $y$ would be an integer or some other non-excessively long number?
Example, $y=\log_{\frac{3}{10}}x$
I can get the key point $(\frac{3}{10},1)$ and the x-intercept $(1,0)$ but not a third point.
Is this possible?
Alternative approach:
For any $r > 0, r \neq 1, x > 0$ you have that
$$\log_r(x) = \frac{\ln(x)}{\ln(r)}. \tag1 $$
In (1) above, for a fixed value $r$, the RHS denominator of $\ln(r)$ is easily computed using a hand-held or online calculator. Further, you should be able to obtain a copy of the graph of $y = \ln(x)$, either manually, by plotting selecting points via a calculator, considering the varying slope (AKA derivative) of the graph, or finding an online copy of the graph.
Then, using (1) above, assuming that an accurate enough graph of $y = \ln(x)$ is to be construed as known, it is then simply a matter of applying to the graph, the fixed scalar $~\displaystyle \frac{1}{\ln(r)}.$
Note that applying a fixed scalar to an existing $2$ dimensional graph does not alter the shape of the graph. Instead, it is as if the $y$-axis scaling has simply been altered by the scalar.