A logarithm asks "what power do I raise the base to?". Because a logarithm is an exponent, it obeys the same three rules the index laws obey — only mirrored. The most useful one turns a multiplication into an addition:

Splitting a tough product into a tidy sum is exactly what made logarithms a calculating tool for centuries. The other two laws follow the same pattern.

The three laws

Each log law is the shadow of an index law. Multiplying powers adds their exponents, so logging a product adds the logs:

Dividing powers subtracts their exponents, so logging a quotient subtracts the logs:

Raising a power to a power multiplies the exponents, so the exponent inside a log comes out to the front as a multiplier:

Read together: products become sums, quotients become differences, and powers become products. Every law drops the operation down one level.

See it built

Here is why the product law is true, straight from the index law. Write both numbers as powers of the base, multiply (which adds the exponents), then log both sides. Step through it.

See it explained

Sal Khan introduces the logarithm properties and shows where each one comes from.