Exponentials and logarithms

e^x, e≈2.718; ln (natural log) is its inverse. Logs convert exponential equations.

e^x is its own derivative. ln(x) inverts e^x. Log laws: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a. Change of base: log_a b = ln b / ln a.

Worked examples

2^x = 10 ⇒ x = log₂ 10 = ln 10 / ln 2 ≈ 3.32.

e^2x = 5 ⇒ 2x = ln 5 ⇒ x ≈ 0.805.

log₃ 81 = 4 (because 3⁴ = 81).

Frequently asked questions

Why e?

Compound interest as compounding frequency → ∞ gives e. Calculus: derivative of e^x is e^x.

ln vs log?

ln is log base e. log without subscript usually means base 10 (or sometimes base e — check context).