Logarithm Rules - Product, Quotient, and Power

What Are Logarithm Rules?

A logarithm answers the question "to what power must I raise the base to get this number?" If $b^y = x$, then $\log_b(x) = y$. For example, $\log_2(8) = 3$ because $2^3 = 8$.

The logarithm rules are a small set of identities that let you rewrite log expressions in simpler forms. Three rules carry almost all of the work:

That conversion — turning hard operations into easier ones — is exactly why logarithms were invented in the first place.

The Three Core Logarithm Rules

1. The Product Rule

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

In words: The log of a product equals the sum of the logs.

Example: $\log_2(8 \times 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5$. Check: $\log_2(32) = 5$ because $2^5 = 32$. ✓

Why it works: Let $m = \log_b(x)$ and $n = \log_b(y)$. Then $x = b^m$ and $y = b^n$. So $xy = b^m \cdot b^n = b^{m+n}$ (by the exponent product rule). Taking the log of both sides: $\log_b(xy) = m + n = \log_b(x) + \log_b(y)$.

2. The Quotient Rule

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

In words: The log of a quotient equals the log of the numerator minus the log of the denominator.

Example: $\log_2!\left(\frac{16}{4}\right) = \log_2(16) - \log_2(4) = 4 - 2 = 2$. Check: $\log_2(4) = 2$. ✓

Why it works: Same setup as above. If $x = b^m$ and $y = b^n$, then $\frac{x}{y} = b^{m-n}$. So $\log_b!\left(\frac{x}{y}\right) = m - n$.

3. The Power Rule

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

In words: The log of a power equals the exponent times the log of the base.

Example: $\log_2(8^3) = 3 \log_2(8) = 3 \cdot 3 = 9$. Check: $8^3 = 512$ and $\log_2(512) = 9$ because $2^9 = 512$. ✓

Why it works: If $x = b^m$, then $x^n = (b^m)^n = b^{mn}$. So $\log_b(x^n) = mn = n \log_b(x)$.

How Do You Prove the Three Logarithm Rules?

Each of the three core rules has a short proof — and the proofs all rest on the same idea: a logarithm is just the inverse of an exponential, so any log identity is an exponential identity dressed up in different notation.

Proof of the Product Rule

Let $m = \log_b x$ and $n = \log_b y$. By the definition of logarithm: $b^m = x$ and $b^n = y$.

Multiply:

$$xy = b^m \cdot b^n = b^{m+n}$$

Taking $\log_b$ of both sides:

$$\log_b(xy) = m + n = \log_b x + \log_b y \qquad \blacksquare$$

Proof of the Quotient Rule

Same setup: $m = \log_b x$, $n = \log_b y$, so $x = b^m$, $y = b^n$.

Divide:

$$\frac{x}{y} = \frac{b^m}{b^n} = b^{m-n}$$

Taking $\log_b$:

$$\log_b!\left(\frac{x}{y}\right) = m - n = \log_b x - \log_b y \qquad \blacksquare$$

Proof of the Power Rule

Let $m = \log_b x$, so $x = b^m$. Raise to the power $n$:

$$x^n = (b^m)^n = b^{mn}$$

Taking $\log_b$:

$$\log_b(x^n) = mn = n \log_b x \qquad \blacksquare$$

Every log rule reduces to a one-line exponent law. Once a student sees this, the rules stop being three things to memorise and become three faces of one relationship: $\log$ is the inverse of $b^{(\cdot)}$.

Two Supporting Identities

Alongside the three core rules, two additional identities are worth memorising:

$$\log_b(1) = 0 \qquad \log_b(b) = 1$$

These follow directly from the definition: $b^0 = 1$ and $b^1 = b$. They show up constantly when simplifying log expressions.

The Change-of-Base Formula

Calculators usually compute logarithms only in two bases: $\log_{10}$ (the common log) and $\log_e$ (the natural log, $\ln$). To compute a log in any other base, use:

$$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$$

where $c$ is any base you can actually compute — usually 10 or $e$.

Example. Compute $\log_2(50)$.

$$\log_2(50) = \frac{\log_{10}(50)}{\log_{10}(2)} \approx \frac{1.699}{0.301} \approx 5.64$$

Check: $2^{5.64} \approx 50$. ✓

Where Logarithms Appear in the Real World

• Richter scale. A magnitude-7 earthquake releases about $10$ times more energy than a magnitude-6. The scale is $\log_{10}$.

• Decibels. Sound intensity doubles roughly every $3$ dB. A jet engine ($140$ dB) is $10^{14/10} \approx 25{,}000{,}000$ times more intense than ordinary conversation ($60$ dB).

• pH scale. A pH-3 acid is $10$ times more acidic than pH-4. The scale is $-\log_{10}[\text{H}^+]$.

• Stellar magnitudes. Each magnitude step is $\sqrt[5]{100} \approx 2.512$ times the brightness — a logarithmic scale that goes back to Hipparchus in 130 BCE, formalised with logarithms in the 19th century.

• Compound interest and growth. The time required to double your money at rate $r$ is $t = \log(2) / \log(1 + r)$.

• Information theory. Claude Shannon's 1948 definition of information is $\log_2$-based — one bit of information halves the uncertainty.

At Bhanzu, our trainers always teach the three rules alongside one of these real-world examples. The rule is easier to remember once a student has seen that the Richter scale is literally a logarithm in action.

A Worked Example — Wrong Path First

Simplify $\log_2(8 \cdot 4)$.

The intuitive (wrong) approach. A student new to logarithms applies the product rule to the outside of the log instead of the inside.

$$\log_2(8 \cdot 4) \stackrel{?}{=} \log_2(8) \cdot \log_2(4)$$

The result is $3 \cdot 2 = 6$. But $8 \cdot 4 = 32$ and $\log_2(32) = 5$, not $6$. The answer is wrong.

Why it fails. The product rule says $\log_b(xy) = \log_b(x) + \log_b(y)$ — the sum of logs, not the product of logs. The multiplication is inside the log; the addition is outside. The memorizer who learned "the product rule says you can split a product" and forgot the operator switch hits this exactly.

The correct method.

$$\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = \boxed{5}$$

Check: $\log_2(32) = 5$ because $2^5 = 32$. ✓

The conceptual point: the product rule turns multiplication into addition, not multiplication into a different multiplication. Every logarithm rule converts the operator — that's the whole reason logarithms are useful.

Common Mistakes with Logarithm Rules

Mistake 1: Splitting the log of a sum

Where it slips in: Anywhere a student wants to "distribute" a logarithm across an addition.

Don't do this: $\log(x + y) = \log(x) + \log(y)$. This is false.

The correct way: There is no logarithm rule for sums. $\log(x + y)$ stays as $\log(x + y)$ — you cannot simplify it further using logarithm rules. The product rule applies to products inside the log, not sums. The rusher who pattern-matches "split things across" hits this most often.

Mistake 2: Splitting the log of a product as a product of logs

Where it slips in: Mistaking the product rule's right-hand side as $\log(x) \cdot \log(y)$ instead of $\log(x) + \log(y)$.

Don't do this: $\log_2(8 \cdot 4) = \log_2(8) \cdot \log_2(4) = 3 \cdot 2 = 6$.

The correct way: $\log_2(8 \cdot 4) = \log_2(8) + \log_2(4) = 3 + 2 = 5$. The product inside the log becomes a sum outside. The memorizer who skipped the operator-switch step makes this mistake repeatedly.

Mistake 3: Power rule with the wrong exponent location

Where it slips in: $(\log x)^2$ and $\log(x^2)$ look similar but mean different things.

Don't do this: Treating $(\log x)^2$ and $\log(x^2)$ as equal.

The correct way: $\log(x^2) = 2 \log(x)$ — the power is inside the log, so the rule applies. $(\log x)^2$ is just the square of $\log x$ — the exponent is outside, so the rule does not apply. The second-guesser who pauses to ask "is the exponent inside or outside the log?" is asking the right question.

The real-world version of the mistake. In 1865, the British physicist John Tyndall measured atmospheric absorption of heat using carefully calibrated experiments — but early CO₂ measurement instruments had a logarithmic response that some experimenters misread as linear, leading to systematically wrong estimates of climate sensitivity for decades.

The same shape of error as $(\log x)^2$ vs $\log(x^2)$ — confusing where the operation applies — produced real scientific drift. Mathematical precision in logarithm rules isn't pedantry; in measurement-based science, the right rule is what keeps the truth from the close-but-wrong approximation.

The Mathematicians Who Shaped Logarithms

John Napier (1550–1617, Scotland) — Invented logarithms in his 1614 book Mirifici Logarithmorum Canonis Descriptio. His goal was practical: to save astronomers from spending months on multiplications. Napier also invented "Napier's bones" — a manual calculation aid that predates the slide rule.

Henry Briggs (1561–1630, England) — Developed the base-10 logarithm tables in 1617 in collaboration with Napier shortly before Napier's death. Briggs's tables became the standard reference for the next three centuries.

Leonhard Euler (1707–1783, Switzerland) — Established the deep connection between logarithms, exponentials, and the number $e$. Euler's identity $e^{i\pi} + 1 = 0$ links the natural logarithm to imaginary numbers and remains one of the most celebrated equations in mathematics.

Three mathematicians, two centuries, one tool that "doubled the life of the astronomer."

A Practical Next Step

Try these three problems before moving to logarithmic equations and the natural log.

1. Simplify $\log_2(16 \cdot 8)$ using the product rule.

2. Simplify $\log_3(81 / 9)$ using the quotient rule.

3. Compute $\log_5(125)$ — without a calculator. (Hint: rewrite $125$ as a power of $5$.)

If you found problem 3 confusing, use the definition: $\log_5(125)$ asks "what power of 5 gives 125?" and $5^3 = 125$. Want a live Bhanzu trainer to walk through more log problems? Book a free demo class — online globally.