Logarithm Calculator

Evaluate logs in any base, apply the log laws, and solve logarithmic and exponential equations.

Try an example:

What a logarithm is

A logarithm answers the question "what power do I raise the base to?". Because $2^{6} = 64$ , we write $lo g_{2} 64 = 6$ . In general:

lo g_{b} a = c ⟺ b^{c} = a

$ln$ is the natural logarithm, with base $e \approx 2.718$ , and $lo g$ with no base usually means base 10.

Type or paste your problem into the box above — for example log_2(64). Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

Use the log laws to combine everything into a single logarithm on each side.
Convert to exponential form — or, for an exponential equation, take logs of both sides.
Solve, then reject any answer that would take the log of zero or a negative number.

Combine the logsBy the product law, $lo g_{2} (x (x - 2)) = 3$ .
Convert to exponential form $x (x - 2) = 2^{3} = 8$
Solve the quadratic $x^{2} - 2 x - 8 = 0$ , so $(x - 4) (x + 2) = 0$ and $x = 4$ or $x = - 2$ .
Check the domain $lo g_{2} (- 2)$ is undefined, so $x = - 2$ is rejected. The solution is $x = 4$ , and indeed $lo g_{2} 4 + lo g_{2} 2 = 2 + 1 = 3$ .

$lo g (x + y)$ is not $lo g x + lo g y$ . The product law applies to $lo g (x y)$ only.

Always check answers in the original equation. The log of zero or of a negative number is undefined.

$lo g x^{2} = 2 lo g x$ for $x > 0$ , but $(lo g x)^{2}$ is something different.

How do I type a log with a base?

Use an underscore: log_2(8) for $lo g_{2} 8$ . ln(x) is the natural log, and log(x) with no base means base 10.

Can it solve exponential equations?

Yes. For $3^{2 x} = 81$ it rewrites 81 as $3^{4}$ and solves $2 x = 4$ . When the bases don't match, it takes logarithms of both sides.

Can it condense log expressions?

Yes. Ask it to condense 2 log x - log y and it applies the power and quotient laws in reverse to get $lo g \frac{x ^{2}}{y}$ .

What's the difference between log and ln?

Only the base: $ln$ uses $e$ and $lo g$ usually uses 10. The change-of-base formula converts between them.