Common and Natural Logarithms: Explained with Examples

The two logarithms you will meet most often are the common logarithm, written log x, which means base 10, and the natural logarithm, written ln x, which means base e, where e ≈ 2.71828. When no base is written in school mathematics, base 10 is understood.

They matter because base 10 matches our decimal counting and scientific notation, while base e matches how nature grows — continuously compounding quantities like populations, investments, and radioactive atoms all speak in ln.

Key Concepts

• Common logarithm: log x = log_10 x; log 1000 = 3 because 10³ = 1000.
• Natural logarithm: ln x = log_e x; ln e = 1 because e¹ = e.
• The constant e ≈ 2.71828: an irrational number arising from continuous growth, not a rounded convenience.
• Converting between them: ln x = (log x)/(log e) ≈ 2.3026 · log x, and log x = (ln x)/(ln 10) ≈ 0.4343 · ln x.
• Notation convention: a bare log means base 10 in school mathematics (in some advanced texts it means e — context decides).

Formulas

Both are ordinary logarithms with fixed bases, so every log law applies to them. The conversion formulas below are just the change-of-base formula with c = 10 or c = e.

• log x = log_10 x and ln x = log_e x
• ln x = (log x)/(log e) ≈ 2.3026 · log x
• log x = (ln x)/(ln 10) ≈ 0.4343 · ln x
• log e ≈ 0.4343 and ln 10 ≈ 2.3026

Worked Example

Evaluate ln 10 and log(e²). For ln 10, use the standard value ln 10 ≈ 2.3026. Verify: e^2.3026 = e^2.3 × e^0.0026 ≈ 9.9749 × 1.0026 ≈ 10.000, confirming it. For log(e²), apply the power rule: log(e²) = 2 · log e ≈ 2 × 0.4343 = 0.8686. Verify: 10^0.8686 = e^(0.8686 × 2.3026) = e^2.0000 ≈ 7.389, and e² ≈ 7.3891 — they match.

Common Misconceptions

• "ln means base 10." ln is always base e; the plain log is the base-10 one.
• Treating a baseless log as base e in school work. In school mathematics log without a base means base 10.
• "e is just 2.7 rounded." It is the irrational constant 2.71828…, defined by continuous growth as the limit of (1 + 1/n)ⁿ.

Real-World Applications

• pH in chemistry uses common logarithms: pH = −log[H⁺].
• Continuous compound interest and half-life formulas use ln.
• Calculus: the derivative of ln x is the beautifully simple 1/x.
• Decibel and Richter scales are built on common logarithms.

Common logarithms (base 10, written log) fit decimal measurement; natural logarithms (base e, written ln) fit continuous growth. Convert with ln x ≈ 2.3026 · log x, and always check which base a bare "log" means in your context.