Logarithm & Exponent Calculator

What Is a Logarithm?

A logarithm is the inverse operation of an exponent. While an exponent asks "what do I get if I multiply a base by itself repeatedly?", a logarithm asks the reverse question: "what power must I raise the base to, to get this number?" For example, since 2³ = 8, the log base 2 of 8 is 3, written log₂(8) = 3. This calculator covers three of the most common logarithm types people search for: the log base 10 calculator (common log, no base shown), the natural log calculator (ln, base e), and the log base 2 calculator (binary log, common in computer science). For a slower, from-scratch introduction to the concept, Khan Academy's logarithm lessons are a solid starting point.

An exponent (or power) tells you how many times to multiply a base by itself — for example 2⁴ = 2×2×2×2 = 16. This calculator's exponent mode handles positive whole-number exponents, negative exponents, and fractional exponents, all explained further down this page.

Formula Explained

A logarithm answers: "to what power must the base be raised to get this number?"

• log (no base shown) = log base 10
• ln = natural log, base e ≈ 2.71828
• Change of base: log_b(x) = ln(x) / ln(b) — this is how any base can be computed using only ln or log10
• Key rule: logarithms and exponents are inverse operations of each other

Product rule: log(ab) = log(a) + log(b)  |  Quotient rule: log(a/b) = log(a) − log(b)  |  Power rule: log(aⁿ) = n·log(a)

Laws of Exponents

These rules apply whenever the bases match:

Example: 5² × 5³ = 5⁵ = 3,125, since (5×5) × (5×5×5) is five copies of 5 multiplied together.

Negative and Fractional Exponents

Negative Exponents

A negative exponent means "take the reciprocal, then raise to the positive exponent."

Example: 5⁻⁴ = 1/5⁴ = 1/625 = 0.0016

Fractional (Rational) Exponents

A fractional exponent represents a root. The denominator of the fraction is the root to take.

Example: 16^(1/2) = √16 = 4. Example: 8^(2/3) = (³√8)² = 2² = 4.

Worked Examples — Step by Step

Example 1: log₂(32)

1. Calculate log₂(32) — "2 raised to what power gives 32?"
2. Try powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32
3. 2⁵ = 32, so log₂(32) = 5
4. Verify: 2⁵ = 32 ✓

Example 2: Negative and Fractional Exponents

(a) Negative exponent — 5⁻⁴

1. A negative exponent means take the reciprocal first: 5⁻⁴ = 1/5⁴
2. Calculate 5⁴ = 5 × 5 × 5 × 5 = 625
3. Take the reciprocal: 1/625 = 0.0016

(b) Fractional exponent — 16^0.5

1. An exponent of 0.5 (= 1/2) means take the square root
2. 16^0.5 = √16
3. √16 = 4, since 4 × 4 = 16 — the same result you'd get from a square root calculator

Example 3: Bacteria Doubling (Real-World Exponential Growth)

A bacteria culture doubles every 20 minutes. Starting with 1 bacterium, how many are there after 2 hours?

1. 2 hours = 120 minutes. Number of 20-minute intervals: 120 ÷ 20 = 6
2. Each interval doubles the population, so this is 2 raised to the 6th power
3. 2⁶ = 2×2×2×2×2×2 = 64

This is why exponential growth feels deceptively slow at first, then explodes: after just 10 hours (30 intervals), the same culture would reach 2³⁰ ≈ 1.07 billion bacteria — a number large enough that scientists would typically write it in scientific notation (roughly 1.07 × 10⁹).

Frequently Asked Questions

What is a logarithm in simple terms?

A logarithm answers: 'to what power must the base be raised to get this number?' log₂(8) = 3 because 2³ = 8.

What is the difference between log and ln?

log (common log) uses base 10. ln (natural log) uses base e ≈ 2.718. Both are used in science and mathematics.

What is Euler's number (e)?

e ≈ 2.71828 is a mathematical constant. It is the base of natural logarithms and appears in growth and decay formulas.

How do I use the change of base formula?

log_b(x) = log(x) / log(b). This lets you calculate any base using your calculator's log or ln button.

What is an antilogarithm?

The inverse of a logarithm. If log₁₀(x) = 3, then the antilog is 10³ = 1000. It reverses the logarithm operation.

Why is log(0) undefined?

Because no power of any positive base can equal zero. As x approaches 0 from the right, log(x) approaches negative infinity.

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