Logarithm Calculator
Natural log (ln), common log (log₁₀), binary log, or any base—with steps.
Logarithm Calculator
Calculate logarithms with any base including natural log (ln) and common log (log₁₀)
Quick Examples
What a logarithm is
logb(x) = y means by = x: the exponent that turns base b into x. Defined for x > 0; base must be positive and not 1.
Napier introduced logs in the early 1600s to speed multiplication. Today they still turn products into sums and show up in growth, scales, and algorithms.
ln: base e ≈ 2.718 (calculus, continuous growth)
log / log₁₀: base 10 (pH, orders of magnitude)
log₂: base 2 (bits, binary search depth)
Rules
logb(x) = logc(x) / logc(b)
Change of base: usually ln(x)/ln(b) or log₁₀(x)/log₁₀(b) on a calculator.
Where logs show up
- pH = −log₁₀[H⁺]; dB = 10 log₁₀(P₁/P₀); Richter uses log amplitude
- Finance: continuous compounding and log returns ln(P₁/P₀)
- CS: O(log n) algorithms, entropy in bits (log₂), tree height
- Solving: if logb(x) = c then x = bc; combine logs before converting
Always check domain after algebra—arguments must stay positive, or you pick up extraneous roots.
Frequently Asked Questions
Why no log of negatives?
No real power of a positive base is negative. Complex logs exist but are multi-valued; stick to x > 0 for real calc.
ln vs log?
ln is always base e. Bare “log” means base 10 in many applied fields and natural log in some math texts—check the context.
Why “natural” log?
Because d/dx ln(x) = 1/x and ex is its own derivative—handy in calculus and continuous growth models.
Logs and exponentials?
Inverses: b^(logb(x)) = x and logb(bx) = x.
Fractional or negative bases?
Bases between 0 and 1 are allowed (decreasing function). Negative bases are avoided in elementary real math.
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