Below you will find an example of the kind of calculator they DON'T want you to use. You will notice the $\log_ \square \square $ button just below the "ON" button. This allows you to calculate things like $\log_9 (27) $ directly.
The problem with this calculator is that it allows you to get "correct answers" without having gained any understanding of what logarithms are. This is not an issue if the goal is to create machines that can mindlessly churn out answers, but I guess that the goal in your case is to gain a deeper understanding than that.
So, let's consider $\log_9 (27) $ for amoment. The calculator below will tell you that the answer is $1.5$, but why is that?
The key (in my opinion) is to remember that logarithms are powers.
Asking "What is the logarithm of $27$?" is equivalent to asking "What power gives the result $27$?" and of course depends on us having an agreed base.
Here I have stated that I want the logarithm of $27$ with the base equal to $9$. This means that I want to solve the equation:
The first reaction may be that this is not obvious. After all, the only easy ways to express $27$ as a power are these:
and none of these has $9$ as the base.
However, we do know that $3=9^\frac 12$, so we can rewrite the third of those expressions as:
$$27=\left(9^\frac 12 \right)^3=9^\frac 32$$
Now in answer to the question "What is the logarith of 27 (with base 9)" we can ask ourselves "What power of 9 is 27?" and confidently state "$\frac 32$" or "$1.5$"