Two logarithm buttons, one calculator. Press log and you get base-10. Press ln and you get base-e. Why both, and when do you use which? Here's the practical answer for U.S. math students.

The two logs at a glance

  • log(x) = log₁₀(x) — the "common log," base 10
  • ln(x) = loge(x) — the "natural log," base e ≈ 2.71828

Both answer the same question: "What power do I raise the base to in order to get x?"

  • log(1000) = 3, because 10³ = 1000.
  • ln(e) = 1, because e¹ = e.

Why base 10?

Common log uses base 10 because we use a base-10 number system. log(x) tells you roughly how many digits x has — log(100) = 2, log(10,000) = 4. Useful for orders-of-magnitude thinking and for scales like decibels, pH, the Richter scale, and stellar magnitude — all engineered around base 10 because humans think in powers of 10.

Why base e?

Natural log shows up in calculus and any model of continuous growth or decay. The function ex has the unique property that its derivative equals itself, making it the "natural" choice for differential equations, compound interest at infinitely-fast compounding, radioactive decay, population growth models, and probability distributions.

The number e ≈ 2.71828 is irrational and transcendental — like π, it has no exact decimal expansion. It arises from the limit (1 + 1/n)n as n → ∞, which is what continuous compounding converges to.

Where you'll see common log

  • pH: pH = −log[H⁺]. A pH of 4 has 10× more hydrogen ions than pH 5.
  • Decibels: dB = 10 log(P₁/P₀). 60 dB is 1,000,000× the reference power.
  • Richter scale: magnitude differences are powers of 10. M7 releases ~32× the energy of M6.
  • Stellar magnitude: a 5-magnitude difference = 100× the brightness.
  • SAT/ACT problems: log appears mainly when the numbers are powers of 10.

Where you'll see natural log

  • Continuous compound interest: A = Pe^(rt). To solve for time: t = ln(A/P) / r.
  • Radioactive decay: N(t) = N₀ × e^(−λt). Half-life: t = ln(2) / λ ≈ 0.693 / λ.
  • Population growth: P(t) = P₀ × e^(rt) for unlimited resources; logistic model otherwise.
  • Calculus integrals: ∫(1/x) dx = ln|x| + C. Natural log appears in any integral involving 1/x.
  • AP Calc and college math: ln dominates. Switch your default away from log.

The change-of-base formula

If your calculator only has log (or only has ln) but you need a different base, use:

logb(x) = log(x) / log(b) = ln(x) / ln(b)

Both versions give the same answer. Examples:

  • log₂(50) = log(50) / log(2) ≈ 1.699 / 0.301 ≈ 5.644
  • log₂(50) = ln(50) / ln(2) ≈ 3.912 / 0.693 ≈ 5.644 ✓

The TI-84 has a logBASE function (MATH → A in newer firmware) that does this automatically, but the change-of-base formula always works.

Algebraic identities (work for any base)

  • log(ab) = log(a) + log(b)
  • log(a/b) = log(a) − log(b)
  • log(an) = n × log(a)
  • log(1) = 0 (any base)
  • logb(b) = 1

These let you simplify log expressions before evaluating. log(8) = log(2³) = 3 log(2), so if you know log(2) ≈ 0.301, you know log(8) ≈ 0.903 without further calculation.

Common mistakes

  • log(a + b) ≠ log(a) + log(b). The product rule applies to multiplication, not addition. There is no shortcut for log of a sum.
  • Confusing log and ln in calculus. The derivative of ln(x) is 1/x. The derivative of log(x) is 1/(x × ln(10)) — there's a hidden conversion factor.
  • Forgetting the domain. log(x) and ln(x) are only defined for x > 0. log(0) and log(negative) are undefined.

Solving exponential equations

Logarithms are the inverse of exponents. To solve bx = k, take the log of both sides:

x = logb(k) = log(k) / log(b)

Worked example: solve 5x = 200.

  • x = log(200) / log(5)
  • x ≈ 2.301 / 0.699 ≈ 3.292
  • Verify: 53.292 ≈ 200. ✓

This pattern shows up on the SAT in problems about half-life, doubling time, and continuous growth. The logarithm is the tool that gets the variable out of the exponent.

Slide rules: logs in your hand

Before the pocket calculator (1970s), engineers used slide rules — analog computing devices based entirely on logarithms. By laying log scales next to each other, you could multiply and divide by sliding pieces. Multiplication of two numbers became addition of their lengths on the log scale.

NASA put men on the moon with slide rules. The Apollo mission flight engineers carried Pickett slide rules; the design relied on log-scale arithmetic the entire way. The slide rule disappeared in five years once electronic calculators became affordable, but the underlying log idea is the same.

Use the calculator

Our logarithm calculator handles any base — common log, natural log, log base 2 (binary log, common in computer science), or any custom base. It also shows the value in all three standard bases simultaneously, so you can spot conversions at a glance.