Laws of Logarithms

Imagine you had to work out 128 \times 512 with no calculator. Slow, fiddly, easy to slip. Now imagine a trick that turns that hard multiplication into an easy addition. That trick is the first law of logarithms — and it is why, for three centuries, every serious calculation on Earth passed through a table of logs.

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 one level down. The headline law turns a product into a sum:

\log_b(xy) = \log_b x + \log_b y

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:

\log_b(xy) = \log_b x + \log_b y

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

\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y

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

\log_b(x^n) = n\,\log_b x

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

For a base b > 0 and positive x, y:

Product law: \log_b(xy) = \log_b x + \log_b y
Quotient law: \log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y
Power law: \log_b(x^n) = n\,\log_b x

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.

Worked examples

Example 1 — build a log from smaller ones. Since 6 = 2 \times 3, the product law splits it apart:

\log 6 = \log(2 \times 3) = \log 2 + \log 3

So if you know \log 2 \approx 0.301 and \log 3 \approx 0.477, you get \log 6 \approx 0.778 by adding — no new lookup needed.

Example 2 — expand a messy expression. Break the product and quotient apart, then bring the power down with the power law:

\log\!\left(\frac{x^2}{y}\right) = \log(x^2) - \log y = 2\log x - \log y

Example 3 — combine into a single log. Run the laws backwards. The multiplier becomes a power, the sum becomes a product, the difference becomes a quotient:

3\log x + \log y - \log z = \log\!\left(\frac{x^3\,y}{z}\right)

Example 4 — solve a log equation. Combine the two logs into one, then convert to index form:

\log_2 x + \log_2 3 = 5 \;\Longrightarrow\; \log_2(3x) = 5 \;\Longrightarrow\; 3x = 2^5 = 32 \;\Longrightarrow\; x = \tfrac{32}{3}

The trick is always the same: squeeze everything into one logarithm, then undo it with the definition.

Each log law mirrors an index law

The log laws aren't three random facts to memorise — they are the index laws seen in a mirror. Because a logarithm is an exponent, whatever the index laws do to exponents, the log laws do to whole numbers, one level down:

b^m \cdot b^n = b^{m+n} \quad\longleftrightarrow\quad \log_b(xy) = \log_b x + \log_b y

\frac{b^m}{b^n} = b^{m-n} \quad\longleftrightarrow\quad \log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y

\left(b^m\right)^n = b^{mn} \quad\longleftrightarrow\quad \log_b(x^n) = n\,\log_b x

Read down the left column: multiplying, dividing, raising to a power. Read down the right: those same three operations, each softened by one step into adding, subtracting, and multiplying. That softening is the whole magic — it is what let a table of logs replace a lifetime of longhand arithmetic.

See it explained

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

Two mistakes lose more marks than any others in this topic:

\log(x + y) is NOT \log x + \log y. The product law is about a product inside the log — \log(xy) = \log x + \log y — not a sum inside. There is simply no law for the log of a sum; \log(x + y) cannot be broken up at all. This is the single most common error in the whole subject, so read the inside of the bracket carefully every time.
\log(x^n) = n\log x is not the same as (\log x)^n. The power law brings the exponent down to the front as a multiplier — it only works when the power is inside the log, on the number itself. \log(x^2) equals 2\log x; but (\log x)^2 means "log x, then square the result" — a completely different value.

For over 300 years, before electronic calculators, the fastest way to multiply two big numbers was to add their logarithms. The product law — \log(xy) = \log x + \log y — is the whole reason that works: look up two logs in a table, add them (easy), and look the total back up. Engineers wore this trick on their belts as the slide rule, a ruler with logarithmic scales that multiplies by sliding one strip past another. NASA's Apollo engineers used slide rules to design the rockets that reached the Moon.

The power law hides an even bigger prize. When the unknown is stuck up in an exponent — "how long until my money doubles?", so 2 = 1.05^{\,t} — the power law reaches up and pulls t back down to ground level where you can solve for it: \log 2 = t \log 1.05. That single move is how you crack exponential equations, from compound interest to radioactive decay. Three little laws, mirroring the index laws perfectly — a genuinely beautiful piece of symmetry.