A logarithm is the inverse of a power: log_b(x) answers "to what power must the base b be raised to get x?". So log₂(8) = 3 because 2³ = 8. Any base can be computed from the change-of-base rule, log_b(x) = ln(x) / ln(b), which is how calculators handle custom bases. The three everyday bases are e (natural log, ln), 10 (common log) and 2 (binary log). Logs turn multiplication into addition and powers into multiplication, which is their enduring usefulness.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the change-of-base rule and logarithm laws.
The logarithm rules
Because the logarithm undoes exponentiation, log_b(b) = 1 and log_b(1) = 0 for any valid base. The change-of-base rule lets you compute a log to any base from natural or common logs — divide the log of the number by the log of the base. The product, quotient and power laws step every operation down a level: products become sums, quotients become differences, and exponents come out front as multipliers, which is exactly what makes logs so handy for simplifying.
Worked example — log base 2 of 8
Scenario: compute log₂(8).
Both routes give 3: 2 raised to the power 3 is 8. The change-of-base check uses ln(8) ≈ 2.0794 and ln(2) ≈ 0.6931, whose ratio is exactly 3. For the same number 8, the other standard logs are ln(8) ≈ 2.0794 and log₁₀(8) ≈ 0.9031. Clean powers of the base give whole-number logs — log₁₀(1000) = 3 and log₁₀(100) = 2 — while ln(e) = 1 because e is the natural base raised to the first power.
Frequently Asked Questions
The power a base needs to reach a number. log₂(8) = 3 since 2³ = 8. Inverse of exponents.
Change of base: log_b(x) = ln(x)/ln(b). log₂(8) = 2.0794/0.6931 = 3.
Bases e, 10 and 2. For 8: ln ≈ 2.0794, log₁₀ ≈ 0.9031, log₂ = 3.
log(xy)=log x+log y; log(x/y)=log x−log y; log(xⁿ)=n log x.
Undefined for reals — bˣ is always positive. Enter x > 0.