Logarithm Calculator
Evaluate any logarithm instantly — log base b, natural log (ln) and common log (log₁₀). Pro mode adds solve-for-x, expression condenser, log-to-exponential converter, derivative calculator and full step-by-step working.
Convert between logarithmic form logb(x) = y and exponential form by = x. Enter any two values to find the third.
Apply the change-of-base formula to rewrite any logarithm in terms of log₁₀ or ln. Useful when your calculator only has log₁₀ and ln buttons.
Place the unknown x in any position — in the argument, the base, or the result. The solver finds x algebraically and shows every step.
Enter a multi-term log expression and condense it into a single logarithm using the Product, Quotient and Power rules. Use + for product, - for quotient, and a leading coefficient for power. Example: 2*log(a)+log(b)-log(c)
Find the derivative d/dx of a logarithmic function. Enter the function and base. Supports ln(x), log(x), log_b(x), and functions of x inside the log.
Free Logarithm Calculator — Any Base, With Steps
A logarithm calculator should do more than spit out a number — it should show you how it got there, especially if you're using it to check homework. This tool evaluates any logarithm you throw at it: common log (base 10), natural log (ln, base e), or any custom base, then shows the exponential form alongside the result so you can see exactly what the answer means. Pro Mode adds full step-by-step working, matching what you'd write out by hand.
If you're specifically searching for a common logarithm calculator or a logarithm scientific calculator-style tool, the Evaluate panel covers exactly that — pick the log₁₀ preset, enter your number, and get the result instantly alongside natural log and log base 2 as quick reference values.
What this tool actually does: evaluates logarithms in any base, converts between logarithmic and exponential form, and applies the change-of-base formula. Pro Mode adds solving for an unknown x in any position, condensing multi-term expressions into a single logarithm, and differentiating logarithmic functions — each with full step-by-step working.
How to Calculate a Logarithm — Step by Step
How to Do a Logarithm on a Calculator
Most physical scientific calculators only have two logarithm buttons — log for base 10 and ln for base e. If you need a logarithm in any other base, like log base 5 or log base 2, your calculator can't do it directly with a single button press. The workaround is the change-of-base formula: logb(x) = log(x) ÷ log(b), computed using the log or ln button twice and then dividing. This tool's Change of Base panel applies that formula automatically and shows the numerator and denominator separately, so you can see exactly what your physical calculator would need to compute.
Solving a Logarithm for x
Where x actually sits changes how you solve for it. If x is the argument — logb(x) = y — convert to exponential form and compute x = by. If x is the base — logx(n) = y — solve x = n(1/y). If x is the result itself — logb(n) = x — you're just evaluating the logarithm directly. Pro Mode's Solve for x panel handles all three cases, with the correct algebraic rearrangement and full working shown for each.
Logarithm Laws Reference
These laws work in reverse too — Pro Mode's expression condenser takes a multi-term expression like 2·log(a) + log(b) − log(c) and combines it into a single logarithm, log(a²·b / c), applying the Power, Product and Quotient rules in the correct order and showing each step.
A base-10 logarithm, usually written log(x). It asks: 10 to what power equals x?
log is base 10, ln is base e (≈2.71828) — both are logarithms, just different bases.
Yes, in Pro Mode — whether x is the argument, the base, or the result.
Yes, fully free, including Pro Mode's solve-for-x, condenser and derivative tools.
Frequently Asked Questions
Common questions about logarithms, calculators and step-by-step working
A common logarithm is a logarithm with base 10, usually written as log(x) without a visible base, or explicitly as log10(x). It answers: 10 raised to what power equals x? This calculator's log₁₀ preset evaluates common logarithms directly.
Most scientific calculators have log (base 10) and ln (base e) buttons. For any other base, use the change-of-base formula: log_b(x) = log(x) / log(b). This tool applies that formula automatically for any base you enter.
Depends where x sits. Argument (log_b(x) = y): x = b^y. Base (log_x(n) = y): x = n^(1/y). Result (log_b(n) = x): evaluate directly. Pro Mode's Solve for x panel handles all three and shows the working.
Use the log laws: Product Rule combines added logs, Quotient Rule combines subtracted logs, Power Rule moves a coefficient inside as an exponent. E.g. 2·log(a) + log(b) − log(c) condenses to log(a²·b/c). Pro Mode's condenser does this automatically.
log usually refers to the common logarithm (base 10), while ln refers to the natural logarithm (base e, ≈2.71828). Both are logarithms — different bases give different results for the same number.
Yes, fully free, including the log-to-exponential converter and change-of-base tool. Pro Mode — also free — adds solve-for-x, expression condenser, derivative calculator, step-by-step working, history and a log laws reference.