Logarithm Evaluator

Logarithm Setup

log₂(8) = ?

Base (b)

Argument (x)

Computes log_b(x) = ln(x)/ln(b). Use b=10 for common log, b=e (2.71828…) for ln.

Examples

log₂(x) = 3 → x = ?

Base (b)

Result (y)

Given log_b(x) = y, find x = b^y.

Examples

log_b(8) = 3 → b = ?

Argument (x)

Result (y)

Given log_b(x) = y, find b = x^(1/y). Result y cannot be 0.

Examples

Result

Enter values above and press Calculate to see the logarithm result, exact form, and an inverse verification.

Answer

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Log Equation

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Inverse Check

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Logarithm Definition

log_b(x) = y means b^y = x

Reading a log: log_b(x) asks "to what power must I raise b to get x?" The base b must be positive and not equal to 1. The argument x must be positive.

Three special logs:

Notation	Base	Meaning
log(x)	10	Common logarithm
ln(x)	e ≈ 2.718	Natural logarithm
log₂(x)	2	Binary logarithm

If log_b(x) comes out to a clean integer, b^integer = x exactly — no rounding needed.

Change of Base Formula

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

Any calculator or language with log base 10 or ln can evaluate any logarithm using this ratio. The base you pick for the ratio cancels — the answer is always the same.

Inverse relationship: Logarithms and exponents undo each other:

b^(log_b(x)) = x and log_b(b^y) = y

This is why the inverse check always works: if log_b(x) = y, then b^y must equal x.

Base b must be positive and b ≠ 1.
Argument x must be positive (log of 0 or negatives is undefined).
log_b(1) = 0 for any base (b^0 = 1).
log_b(b) = 1 for any base (b^1 = b).

Logarithms still feel mysterious?