Properties of Logarithms - Laws, Formulas, Proofs

What Is a Logarithm? (Quick Refresher)

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

The two most common bases are:

• Common logarithm — base 10, often written as $\log(x)$ with no subscript

• Natural logarithm — base $e \approx 2.71828$, written $\ln(x)$

Every property below holds for any base $b$ (with $b > 0$ and $b \neq 1$).

What Are the Properties of Logarithms?

The seven core properties, in compact form:

How Do the Three Main Properties Work? (With Proofs)

Product Rule

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

Proof. Let $m = \log_b x$ and $n = \log_b y$. Then $x = b^m$ and $y = b^n$, so:

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

Taking $\log_b$ of both sides: $\log_b(xy) = m + n = \log_b x + \log_b y$. ∎

Example. $\log_2(8 \times 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5$.

Quotient Rule

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

Proof. Same setup as above: $x/y = b^m/b^n = b^{m-n}$, so $\log_b(x/y) = m - n$. ∎

Example. $\log_3(81/9) = \log_3 81 - \log_3 9 = 4 - 2 = 2$.

Power Rule

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

Proof. Let $m = \log_b x$, so $x = b^m$. Then $x^n = (b^m)^n = b^{mn}$, giving $\log_b(x^n) = mn = n \log_b x$. ∎

Example. $\log_2(8^3) = 3 \log_2 8 = 3 \times 3 = 9$.

How Do You Use the Change-of-Base Formula?

Calculators only compute $\log$ (base 10) and $\ln$ (base $e$). For any other base, use the change-of-base formula:

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

where $c$ is any base your calculator supports.

Example. Compute $\log_5(125)$ using base-10 logs.

$$\log_5(125) = \frac{\log 125}{\log 5} = \frac{2.0969}{0.6990} \approx 3$$

Check: $5^3 = 125$ ✓.

Special case — change to natural log:

$$\log_b x = \frac{\ln x}{\ln b}$$

Often the cleaner form for higher-math contexts where $\ln$ is the more useful base.

What Are the Three Special-Value Properties?

These follow directly from the definition of a logarithm.

Log of 1 = 0

$$\log_b 1 = 0 \qquad \text{(any base } b > 0, b \neq 1\text{)}$$

Because $b^0 = 1$ for any non-zero $b$.

Log of the Base Itself = 1

$$\log_b b = 1$$

Because $b^1 = b$.

Log of a Power of the Base = the Exponent

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

This is the inverse-function property — logarithm undoes exponentiation. For natural log: $\ln(e^n) = n$, and $e^{\ln x} = x$.

What Are the Properties of the Natural Logarithm (ln)?

The natural logarithm $\ln(x)$ is just $\log_e(x)$ — the logarithm with base $e \approx 2.71828$. Every property above carries over verbatim with $e$ as the base; we restate them here because $\ln$ is the form you'll meet most often in calculus, physics, and finance.

Calculus identity (bonus). The natural log is the unique logarithm whose derivative is $1/x$:

$$\frac{d}{dx}\ln x = \frac{1}{x}, \quad x > 0$$

That single property is why $e$ — and therefore $\ln$ — is the "natural" base in pure math and physics. The base-10 log doesn't have a clean derivative; only base $e$ does.

Worked example. Simplify $\ln(e^3 \cdot x^2)$.

$\ln(e^3 \cdot x^2) = \ln e^3 + \ln x^2 = 3 + 2\ln x$.

Worked example — compound interest at continuous rate. A balance growing at continuous annual rate $r$ from initial principal $P_0$ reaches $P(t) = P_0 e^{rt}$. To solve for $t$:

$$t = \frac{1}{r} \ln!\left(\frac{P(t)}{P_0}\right)$$

The natural log is the inverse of continuous-compound growth — which is exactly why finance, biology, and physics use $\ln$, not $\log$.

Why Were Logarithms Invented? (The Real-World GROUND)

"Seeing there is nothing… that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers…" — John Napier, 1614.

Logarithms exist because of a specific 17th-century problem: astronomers and navigators needed to multiply enormous numbers, and multiplication by hand took hours. John Napier — a Scottish mathematician working at Merchiston Castle — published Mirifici Logarithmorum Canonis Descriptio in 1614. His logarithm tables converted multiplication into addition, and his collaborator Henry Briggs built the base-10 tables that became the standard.

The mathematician Pierre-Simon Laplace later wrote that logarithms "by shortening the labours, doubled the life of the astronomer." Calculations that took months took weeks; the impossible became routine.

The slide rule — invented around 1620 — was nothing but a logarithm table on a sliding stick. Slide rules went on the Apollo missions to the moon and stayed standard equipment in every engineering school until cheap electronic calculators arrived in the 1970s.

Today, logarithmic scales describe enormous ranges of natural phenomena:

• The Richter scale — a magnitude-7 earthquake releases 10× the energy of magnitude-6.

• The decibel scale — sound intensity doubles roughly every 3 dB.

• The pH scale — pH 3 is 10× more acidic than pH 4.

• Stellar magnitudes — each step is about $\sqrt[5]{100} \approx 2.512$ times the brightness.

• Information theory — Claude Shannon's 1948 definition of information content uses $\log_2$. One bit halves uncertainty.

A Worked Example

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

The intuitive (wrong) approach. A student applies the product rule incorrectly:

$$\log_2(8 \cdot 4) \stackrel{?}{=} \log_2 8 \cdot \log_2 4 = 3 \cdot 2 = 6$$

Then $6 - \log_2 4 = 6 - 2 = 4$.

But the actual value of $\log_2(8 \cdot 4) - \log_2 4 = \log_2 32 - \log_2 4 = 5 - 2 = 3$, not 4.

Why it fails. The product rule converts multiplication inside the log to addition outside — $\log_b(xy) = \log_b x + \log_b y$, not $\log_b x \cdot \log_b y$. The operator switches; the form doesn't multiply.

The correct method.

$$\log_2(8 \cdot 4) - \log_2 4 = (\log_2 8 + \log_2 4) - \log_2 4 = \log_2 8 = 3$$

Or by quotient rule: $\log_2(8 \cdot 4 / 4) = \log_2 8 = 3$.

The memorizer who learned "split the product" without internalising "sum, not product" hits this mistake repeatedly. At Bhanzu, our trainers teach the conversion as the central idea: every logarithm rule changes the operation level — products become sums, quotients become differences, exponents become coefficients.

What Are the Most Common Mistakes With Properties of Logarithms?

Mistake 1: Splitting $\log(x + y)$ as $\log x + \log y$

Where it slips in: Trying to "distribute" the log over an addition.

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

The correct way: There is no property for the log of a sum. $\log(x + y)$ stays as $\log(x + y)$ — you cannot simplify further using log properties. The rusher who pattern-matches "split things" hits this most.

Mistake 2: Confusing $\log(x^n)$ with $(\log x)^n$

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 exponent is inside the log, so the power rule applies. $(\log x)^2 = \log x \cdot \log x$ — the exponent is outside, applied to the whole logarithm. They are not equal in general. The second-guesser who asks "inside or outside?" is right to ask.

Mistake 3: Treating $\log 0$ as 0

Where it slips in: Recalling that $\log 1 = 0$ and confusing it with $\log 0$.

Don't do this: Writing $\log 0 = 0$.

The correct way: $\log 0$ is undefined — no power of any positive base produces 0. As $x \to 0^+$, $\log_b x \to -\infty$ (it approaches negative infinity). Distinct from $\log 1 = 0$. The memorizer who pattern-matches "log of small number = 0" hits this.

The real-world version of the mistake. In 1865, the British physicist John Tyndall measured atmospheric CO₂ absorption using carefully calibrated equipment — but early CO₂ instruments had a logarithmic response that some experimenters read as linear.

The misreading is the same shape as $\log(x^2)$ vs $(\log x)^2$ — confusing where the operation applies — and it produced systematically wrong estimates of climate sensitivity for decades.

Mathematical precision in log properties isn't pedantry; in measurement-based science, the right property is the difference between truth and plausible drift.

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 multiplication. Also invented "Napier's bones" — a manual calculation aid that predates the slide rule.

Henry Briggs (1561–1630, England) — Developed the base-10 (common) 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 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.

A Practical Next Step

Try these three before moving on to logarithmic equations.

1. Simplify $\log 100 + \log 10$ using the product rule.

2. Simplify $\log_2(64) - \log_2(8)$ using the quotient rule.

3. Compute $\log_3(81)$ — without a calculator.

If problem 3 confused you, use the definition: $\log_3(81)$ asks "what power of 3 gives 81?" and $3^4 = 81$. Want a live Bhanzu trainer to walk through more log problems? Book a free demo class — online globally.