Logarithms

The following is a nice way to think about logarithms:
$\log_b a \equiv$ The number of times $a$ is divided into $b$ parts such that each part becomes $1$.

For example, $1000$ needs to be divided into 10 parts $3$ times to get to 1, so $\log_{10}1000 = 3$.

Reaching arbitrary numbers by repeated division

Now, we may ask—how many times should $a$ be divided into $b$ parts to reach an arbitrary number $x$? We can split $a$ reaching $1$ to $a$ reaching $x$ and then $x$ reaching $1$.

Let $k \equiv$ The number of times $a$ is divided into $b$ parts to reach $x$.

We have,
$\log_b a = k + \log_b x \\ \implies k = \log_b a - \log_b x = \log_b \left (\frac a x \right )$

Intuition for the change of base rule

We will use this to build intuition for the following identity:
$\log_qa = \log_pa . \log_qp$

Informally, the above asks,
how do we relate the number of times we divide $a$ in $q$ parts to reach $1$ and the number of times we divide $a$ in $p$ parts to reach $1$?

Let us look at the first step of calculating $\log_a p$. We divide $a$ into $p$ parts, and each part is of size $a/p$. Just to perform this first step, if I was only allowed to divide by $q$, how many divisions would I need to reach $a/p$ ?

Well, this is the question we asked in the previous section!
The number of divisions is equal to $\log_q \frac{a}{a/p} = \log_q p$. Notice this is independent of $a$, so this holds true to reach any $x/p$ from $x$, in other words, it holds for every step of the calculation of $\log_a p$.

So, we have $\log_q p$ divisions [in the process where you only divide by $q$ to reach $1$] for each of the $\log_p a$ divisions [in the process where you only divide by $p$ to reach $1$].

Which gives us:
$\log_q a = \log_p a. \log_q p$

Yayyy!

I would love to be able to formalize the intuition I have here, and also extend it for other logarithm identities. If you have questions, or anything that may help, feel free to comment, or reach out to me at mahathivempatiresearch@gmail.com.