Logarithm Calculator (log base b of x)

What a logarithm means

A logarithm answers a "what power?" question. $\log_b x = y$ means: raising the base $b$ to the power $y$ gives $x$. It is written the other way in exponent form:

$\log_b x = y \iff b^y = x$

For example, $\log_2 8 = 3$ because $2^3 = 8$ — you have to multiply $2$ by itself $3$ times to reach $8$. A logarithm and an exponent describe the exact same relationship between $b$, $x$, and $y$; the logarithm just solves for the exponent instead of the result.

The logarithm is the inverse of exponentiation

Exponentiation and logarithms undo each other, the same way addition/subtraction and multiplication/division do:

Because they are inverses, raising the base to a logarithm always returns the original value: $b^{\log_b x} = x$. This calculator's "The defining identity" section shows this plugged-in check for every result.

Three named logarithms

Any positive base (other than 1) is valid, but three bases show up so often they have their own names:

Type e in the base field for the natural logarithm — this calculator treats the literal letter e as Euler's number, computed to full precision.

Why x and the base have restrictions

A logarithm is only defined when $x > 0$ and the base $b$ satisfies $b > 0$ and $b \neq 1$:

• $x \leq 0$ is invalid. A positive base raised to any real power always stays positive — there is no real $y$ with $b^y \leq 0$ when $b > 0$.
• $b \leq 0$ is invalid. Raising a negative base to a non-integer power (which almost every logarithm result is) is not a real number.
• $b = 1$ is invalid. $1^y = 1$ for every $y$, so $\log_1 x$ has no single answer — it is left undefined rather than guessed at.

This calculator reports any of those combinations as out of range instead of returning a misleading number.

Exact results vs. decimal approximations

When $x$ is a perfect power of the base — like $\log_2 8$ or $\log_{10} 1000$ — the result is a whole number, computed exactly. Most inputs are not perfect powers, so the result is an irrational decimal, which this calculator rounds to 6 decimal places: $\log_{10} 8 \approx 0.903090$, since no whole power of 10 lands exactly on 8.

Example 1 (exact, binary log): $\log_2 8 = 3$, since $2^3 = 8$.

Example 2 (exact, common log): $\log_{10} 1000 = 3$, since $10^3 = 1000$.

Example 3 (exact, natural log): $\ln e = 1$, since $e^1 = e$.

Example 4 (decimal approximation): $\log_{10} 8 \approx 0.903090$, since $10^{0.903090} \approx 8$.

Where logarithms show up

Logarithms compress huge ranges of numbers into manageable scales — the Richter scale, the decibel scale, and pH are all logarithmic for exactly this reason. In finance, the natural log turns compound growth rates into something you can add and subtract. In computer science, binary logarithms measure how many times a problem can be split in half, which is why sorting and searching algorithms are described in terms of $\log_2 n$. For the reverse question — solving an equation for an unknown exponent — see the related exponent calculators; for whole-number and fractional powers themselves, see the Exponents calculator and the Fractional Exponents calculator.