Q.A scientist running an experiment finds that a particular bacterial colony doubles its population every 20 hours. The experiment starts with 200 bacteria cells. The number of cells is expected to be given by the formula $b = 200(\sqrt{2})^{\frac{t}{20}}$ where $t$ is the number of hours for which the experiment is running. Find the number of hours after which there will be 500 bacteria cells.

What is a logarithm, really?

Before we touch a single law, you need the feel of a logarithm. A logarithm answers one question: "What exponent do I need?"

Take $2^3 = 8$. The exponent is $3$. The logarithm says: "To get $8$ from base $2$, the exponent you need is $3$." We write:

That's it. The logarithm is just the exponent written in a different hat. Every time you see $\log_a b$, read it as: "the exponent on base $a$ that gives $b$".

The three fundamental laws

These are not arbitrary rules. They are direct consequences of the laws of exponents — because logarithms are exponents.

Law 1: Product becomes sum

Why?

Let $\log_a M = x$ and $\log_a N = y$. That means $a^x = M$ and $a^y = N$.

Then $M \cdot N = a^x \cdot a^y = a^{x+y}$.

The exponent that gives $M \cdot N$ is $x+y$, so $\log_a (M \cdot N) = x + y = \log_a M + \log_a N$.

Law 2: Quotient becomes difference

Why?

Same idea: $M/N = a^x / a^y = a^{x-y}$, so the exponent is $x-y$.

Law 3: Power becomes multiplication

Why?

$M^k = (a^x)^k = a^{kx}$, so the exponent is $kx$.

Two special cases you must know

The logarithm of 1

Because $a^0 = 1$ always.

The logarithm of the base itself

Because $a^1 = a$.

Change of base formula

Sometimes you need to switch bases. The formula is:

where $c$ is any positive base (commonly $10$ or $e$).

Why? …