Every core exponent and logarithm rule explained with worked examples, how the two operations relate to each other, and the mistakes that trip up most people.
Published August 31, 2026 · Reviewed by the Ihsabha Editorial Team
Exponents and logarithms show up everywhere from compound interest and population growth to earthquake magnitude and sound volume, and both operations run on a small, fixed set of rules that make otherwise tedious calculations fast once they're memorized correctly. The catch is that a handful of these rules are frequently misapplied — multiplying exponents when they should be added, or misreading what a negative exponent actually does — and a single flipped rule can throw off an entire calculation. This guide walks through every core exponent rule and logarithm rule with worked examples, explains exactly how the two operations are mirror images of each other, and covers the mistakes worth double-checking every time.
An exponent is repeated multiplication written compactly: aⁿ means the base a multiplied by itself n times. So 2⁴ means 2 × 2 × 2 × 2 = 16, not 2 × 4. The base is the number being multiplied, and the exponent (also called a power) is how many times it's multiplied by itself. This single definition is the foundation every exponent rule below is built from — each rule is really just a shortcut for what happens when repeated multiplications of this kind are combined, divided, or nested. This naturally leads to a related question, answered in Reverse Percentage: How to Find the Original Number. You can use the Exponent & Logarithm Calculator to get the result instantly, with no manual math.
When multiplying two powers that share the same base, keep the base and add the exponents: aᵐ × aⁿ = a^(m+n). For example, 2³ × 2⁴ = 2^(3+4) = 2⁷ = 128. This works because 2³ is really (2×2×2) and 2⁴ is (2×2×2×2) — multiplying them together stacks up seven total 2's being multiplied, which is exactly what 2⁷ means. This rule only applies when the bases match; 2³ × 3⁴ cannot be simplified this way, since the two expressions have different bases.
When dividing two powers with the same base, keep the base and subtract the exponents: aᵐ ÷ aⁿ = a^(m−n). For example, 5⁶ ÷ 5² = 5^(6−2) = 5⁴ = 625. Intuitively, dividing cancels out matching factors from the numerator and denominator — 5⁶ ÷ 5² cancels two of the six 5's in the numerator against the two in the denominator, leaving four 5's multiplied together.
When a power is itself raised to another exponent, multiply the exponents: (aᵐ)ⁿ = a^(m×n). For example, (3²)⁴ = 3^(2×4) = 3⁸ = 6,561. This is the rule most frequently confused with the product rule — the key difference is that the product rule adds exponents when multiplying two separate powers, while the power rule multiplies exponents when one power is nested inside another.
Any nonzero number raised to the power of zero equals 1: a⁰ = 1 (for a ≠ 0). This follows directly from the quotient rule: aⁿ ÷ aⁿ = a^(n−n) = a⁰, and any number divided by itself equals 1, so a⁰ must also equal 1. This holds true regardless of how large or small the base is — 1,000,000⁰ and 0.001⁰ both equal exactly 1.
A negative exponent means take the reciprocal of the base raised to the positive version of that exponent: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8. A negative exponent is not the same as a negative number — it doesn't make the result negative, it flips the base into a fraction. This is one of the most commonly confused exponent rules, since it's easy to mistake "negative exponent" for "negative result" at a glance. You may also find it useful to check Markup vs. Margin: How to Calculate Each Percentage Correctly, which covers a related angle.
When a product of two numbers is raised to a power, the exponent applies to each factor separately: (ab)ⁿ = aⁿ × bⁿ. For example, (2×3)² = 2² × 3² = 4 × 9 = 36, which matches directly computing (2×3)² = 6² = 36. This rule is what allows an exponent to be distributed across a multiplication inside parentheses. Use the Exponent & Logarithm Calculator below to check your own numbers quickly and accurately.
The same distribution applies to division: (a/b)ⁿ = aⁿ/bⁿ (for b ≠ 0). For example, (4/2)³ = 4³/2³ = 64/8 = 8, which matches directly computing (4/2)³ = 2³ = 8. Just as with the product rule, the exponent distributes across both the numerator and denominator individually.
A fractional exponent represents a root: a^(1/n) is the same as the n-th root of a. So a^(1/2) is the square root of a, and a^(1/3) is the cube root of a. For a fraction like a^(m/n), the numerator (m) acts as a regular exponent and the denominator (n) acts as a root: a^(m/n) = the n-th root of (aᵐ). For example, 8^(2/3) means the cube root of 8², which is the cube root of 64, equal to 4. This is the bridge that connects the exponent rules covered so far with root operations, since every root can be rewritten as a fractional exponent and manipulated with the same rules.
| Rule | Formula | Example |
|---|---|---|
| Product rule | aᵐ × aⁿ = a^(m+n) | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient rule | aᵐ ÷ aⁿ = a^(m−n) | 5⁶ ÷ 5² = 5⁴ = 625 |
| Power rule | (aᵐ)ⁿ = a^(mn) | (3²)⁴ = 3⁸ = 6,561 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional exponent | a^(1/n) = ⁿ√a | 9^(1/2) = 3 |
A logarithm is the inverse operation of an exponent — it answers a different question about the same three numbers. Where an exponent asks "what do I get if I raise this base to this power?", a logarithm asks the reverse: "what power do I need to raise this base to, to get this number?" Formally, logb(y) = x means exactly the same relationship as bˣ = y — they're two ways of writing the identical fact. For example, since 2³ = 8, it follows that log2(8) = 3: the logarithm of 8, base 2, is 3, because 3 is the power 2 must be raised to in order to produce 8.
The logarithm of a product equals the sum of the logarithms: logb(x × y) = logb(x) + logb(y). For example, log2(4 × 8) = log2(4) + log2(8) = 2 + 3 = 5, which checks out since 4 × 8 = 32 and log2(32) = 5 (because 2⁵ = 32). This rule is the mirror image of the exponent product rule, and it's the exact mechanism that let mathematicians and engineers use logarithm tables and slide rules for centuries to turn tedious multiplication into simple addition, before electronic calculators existed. If you want to go further, How to Calculate a Tip: The Complete Tip Percentage Guide explains this point in more depth.
The logarithm of a quotient equals the difference of the logarithms: logb(x ÷ y) = logb(x) − logb(y). For example, log2(32 ÷ 4) = log2(32) − log2(4) = 5 − 2 = 3, which checks out since 32 ÷ 4 = 8 and log2(8) = 3. Just like its exponent counterpart, this rule turns a division problem into a subtraction problem once everything is expressed in logarithms.
The logarithm of a number raised to a power equals the exponent multiplied by the logarithm of the base number: logb(xⁿ) = n × logb(x). For example, log2(4³) = 3 × log2(4) = 3 × 2 = 6, which checks out since 4³ = 64 and log2(64) = 6 (because 2⁶ = 64). This rule is especially useful for solving equations where the unknown variable sits in an exponent, since it lets that exponent be pulled down in front of the logarithm as an ordinary multiplier.
Two specific bases show up so often that they get their own shorthand notation. A common logarithm uses base 10 and is written simply as "log" with no base shown — log(100) means log10(100), which equals 2. A natural logarithm uses the mathematical constant e (approximately 2.71828) as its base and is written "ln" — ln(x) means loge(x). Base-10 logarithms are common in fields that naturally work in powers of ten, like the Richter scale for earthquakes or the decibel scale for sound, while natural logarithms show up constantly in calculus, compound growth, and any process — like population growth or radioactive decay — that changes continuously rather than in fixed steps. Try the Exponent & Logarithm Calculator to run these numbers with your own figures.
Logarithms exist precisely to solve equations where the unknown sits in the exponent. Take 3ˣ = 81. Taking the logarithm of both sides and applying the power rule: log(3ˣ) = log(81), so x × log(3) = log(81), giving x = log(81) ÷ log(3). Calculating that division gives x = 4 — and checking the answer directly confirms it: 3⁴ = 81. This is the standard method for solving any exponential equation where the base doesn't divide evenly, and it's the same underlying technique used for calculating compound interest doubling time, radioactive half-life, and population growth rates.
Most calculators only compute logarithms directly in base 10 or base e, so finding a logarithm in any other base — like log2(50) — requires converting it first using the change of base formula: logb(x) = log(x) ÷ log(b), using either common (base-10) or natural (base-e) logarithms consistently on both the top and bottom of the fraction. For log2(50), this becomes log(50) ÷ log(2) ≈ 1.699 ÷ 0.301 ≈ 5.644. This formula is what makes every other base calculable using only the "log" and "ln" buttons that appear on a standard calculator, without needing separate functions for every possible base. For a deeper look, our guide on How to Calculate Percentage Off: Discount & Sale Price Formula covers this in more detail.
Logarithms were introduced in the early 1600s, most notably by the Scottish mathematician John Napier, at a time when astronomers, navigators, and engineers routinely needed to multiply and divide enormous numbers by hand — a slow, error-prone process with no mechanical calculators available. Because the logarithm product rule turns multiplication into addition, and the quotient rule turns division into subtraction, a multiplication problem involving two large numbers could be solved by looking up each number's logarithm in a printed table, adding the two logarithms together (a far faster operation than long multiplication), and then looking up which number corresponds to that summed logarithm. This technique, along with the slide rule that mechanized it, remained the standard method for scientific and engineering calculation for roughly 350 years, right up until electronic calculators became widely available in the 1970s made the underlying trick unnecessary for everyday use — even though the mathematical rules themselves are exactly as useful today for solving exponential equations and understanding exponential growth.
Skip the manual rule-tracking. Ihsabha's Exponent & Logarithm Calculator raises any number to a power or finds the logarithm of a number in any base, instantly and accurately.
Exponents govern any process that grows or shrinks by a constant percentage rather than a constant amount — compound interest is the clearest everyday example, since the balance grows by the same percentage each period rather than the same dollar amount, which is exactly what an exponential formula describes. Logarithms, being the inverse, are the natural tool whenever the question flips around from "how much will this grow to?" to "how long will this take to reach a target?" — figuring out how many years it takes an investment to double, or how many half-lives it takes a radioactive sample to decay to a given fraction of its original amount, both require isolating a variable stuck in an exponent, which is exactly the situation logarithms are built to solve. Beyond finance and science, logarithmic scales are also used specifically because human perception itself is roughly logarithmic — the decibel scale for loudness and the Richter scale for earthquake magnitude are both built so that each whole-number increase represents a multiplicative jump in the underlying physical quantity, not a fixed additive one, matching how a listener or a seismograph actually experiences the difference.
When multiplying two powers with the same base, keep the base and add the exponents: aᵐ × aⁿ = a^(m+n). For example, 2³ × 2⁴ = 2⁷ = 128.
A negative exponent means take the reciprocal of the base raised to the positive version of that exponent: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8.
A logarithm is the inverse operation of an exponent. If bˣ = y, then logb(y) = x. The logarithm answers the question: what exponent do I need to raise the base to, to get this number?
The logarithm of a product equals the sum of the logarithms: logb(x × y) = logb(x) + logb(y). This rule is what allowed old slide rules and log tables to turn multiplication into addition.
Exponent rules and logarithm rules mirror each other by design — every exponent rule has a corresponding logarithm rule, because a logarithm is nothing more than the question an exponent answers, asked in reverse. Once the product rule, quotient rule, power rule, and negative/zero exponent rules feel automatic on the exponent side, the matching logarithm rules tend to follow naturally, since they're describing the same underlying relationships from the opposite direction. Keep the base-matching requirement in mind for both operations, double-check whether a rule calls for adding or multiplying exponents, and calculations that once required a page of algebra become a handful of quick, reliable steps.
Work it out instantly with Ihsabha's Percentage Calculator and Percentage Change Calculator. For related reading on Ihsabha's blog, see How to Use a Scientific Calculator: Every Function Explained, Degrees vs. Radians: The Complete Guide to Angle Units and Significant Figures: Rules, Examples & How to Round Correctly.