Log Calculator

One inverse relationship · three ways to solve

Solve any part of logᵦ(x) = y

Find a logarithm, recover its positive argument, or solve for the valid base. The matching exponential form makes every step easier to see and verify.

Natural & custom basesReverse solvingEquivalent forms

log_b(x)?b^?x

Verified logarithm rulesFormula 1.0.0. Calculations retain full precision; answers display up to 12 significant digits.

Inverse exponential relation

Solve one real logarithm relation with every domain condition exposed

The equation log base b of x equals y links three quantities. Selecting the unknown first prevents an inverse calculation from silently crossing the real logarithm's argument and base restrictions.

Side-by-side guide · Inverse exponential relation

Choose which member of the logarithm relation is unknown

Equivalent statement: log_b(x)=y ⇔ b^y=x
Argument domain: x > 0
Base domain: b > 0 and b ≠ 1

Solve for logarithm y
Given positive x and a valid base b, compute y=ln(x)/ln(b). For x=100 and b=10, y=2 because 10 squared is 100.
Solve for argument x
Given b and y, reconstruct x=b^y. A valid real exponential output is positive, matching the logarithm argument domain.
Solve for base b
Given positive x other than 1 and nonzero y, return the positive principal base exp(ln(x)/y). Cases that do not determine a unique valid base are rejected rather than guessed.

Solve one real logarithm relation with every domain condition exposed: The equation log base b of x equals y links three quantities. Selecting the unknown first prevents an inverse calculation from silently crossing the real logarithm's argument and base restrictions.

Choose which member of the logarithm relation is unknown

Solve for logarithm y

Given positive x and a valid base b, compute y=ln(x)/ln(b). For x=100 and b=10, y=2 because 10 squared is 100.

Solve for argument x

Given b and y, reconstruct x=b^y. A valid real exponential output is positive, matching the logarithm argument domain.

Solve for base b

Given positive x other than 1 and nonzero y, return the positive principal base exp(ln(x)/y). Cases that do not determine a unique valid base are rejected rather than guessed.

Verify a base-ten result in exponential form

Treat log_10(0.01)=−2 as an inverse statement rather than a keypad convention.

Validate the inputs
The argument 0.01 is positive, and base 10 is positive and not 1, so the real logarithm exists.
Apply change of base
Compute ln(0.01)/ln(10). The quotient is −2; using a consistent logarithm in numerator and denominator is what matters.
Reverse the relation
Check 10^(−2)=0.01. This exponential reconstruction is the clearest verification and catches a misplaced sign or base.
Plot only when behavior matters
The limit calculator can examine one-sided approach as x moves toward zero, but numerical limiting behavior does not replace the exact inverse relationship.

Distinguish base choice from notation habit

Some disciplines write log for base 10, others for the natural logarithm, and software interfaces vary. This calculator avoids that ambiguity by asking for a numeric base. Enter Math.E numerically when the intended function is ln. Never infer a base from an unlabeled source equation; record the convention before calculation.

Use the exponent relation calculator when the problem is naturally stated as b^y=x, or the equation solver for its supported linear form. This page solves one three-quantity logarithm relation; it does not solve arbitrary equations containing several logs.

Enforce real-domain exclusions before pressing calculate

Zero and negative arguments have no real logarithm. Complex branches are outside this page and must not be inferred from an error message.
Base 1 is invalid because 1 raised to every real exponent remains 1, so it cannot provide a one-to-one inverse relation.
A base between 0 and 1 is valid and produces a decreasing logarithmic function. Do not reject it merely because classroom examples often use bases greater than 1.
Finite-precision output may underflow or overflow at extreme exponents. The workbench rejects non-finite or non-unique inverse cases rather than displaying a plausible but unsupported result.
Record the chosen base alongside every rounded result and inverse check.

Important limitations

This calculator works with finite real numbers. The argument must be positive, the base must be positive and cannot equal one, reverse-base inputs must determine a unique result, and complex or non-finite logarithms are outside its scope.

Review the stated scope, calculation method, assumptions, supported inputs, and effective dates, and obtain qualified advice before using a high-stakes result.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Inverse exponential relation

Solve one real logarithm relation with every domain condition exposed

The equation log base b of x equals y links three quantities. Selecting the unknown first prevents an inverse calculation from silently crossing the real logarithm's argument and base restrictions.

Side-by-side guide · Inverse exponential relation

Choose which member of the logarithm relation is unknown

Equivalent statement: log_b(x)=y ⇔ b^y=x
Argument domain: x > 0
Base domain: b > 0 and b ≠ 1

Solve for logarithm y
Given positive x and a valid base b, compute y=ln(x)/ln(b). For x=100 and b=10, y=2 because 10 squared is 100.
Solve for argument x
Given b and y, reconstruct x=b^y. A valid real exponential output is positive, matching the logarithm argument domain.
Solve for base b
Given positive x other than 1 and nonzero y, return the positive principal base exp(ln(x)/y). Cases that do not determine a unique valid base are rejected rather than guessed.

Choose which member of the logarithm relation is unknown

Solve for logarithm y

Given positive x and a valid base b, compute y=ln(x)/ln(b). For x=100 and b=10, y=2 because 10 squared is 100.

Solve for argument x

Given b and y, reconstruct x=b^y. A valid real exponential output is positive, matching the logarithm argument domain.

Solve for base b

Given positive x other than 1 and nonzero y, return the positive principal base exp(ln(x)/y). Cases that do not determine a unique valid base are rejected rather than guessed.

Verify a base-ten result in exponential form

Treat log_10(0.01)=−2 as an inverse statement rather than a keypad convention.

Validate the inputs
The argument 0.01 is positive, and base 10 is positive and not 1, so the real logarithm exists.
Apply change of base
Compute ln(0.01)/ln(10). The quotient is −2; using a consistent logarithm in numerator and denominator is what matters.
Reverse the relation
Check 10^(−2)=0.01. This exponential reconstruction is the clearest verification and catches a misplaced sign or base.
Plot only when behavior matters
The limit calculator can examine one-sided approach as x moves toward zero, but numerical limiting behavior does not replace the exact inverse relationship.

Distinguish base choice from notation habit

Enforce real-domain exclusions before pressing calculate

Zero and negative arguments have no real logarithm. Complex branches are outside this page and must not be inferred from an error message.
Base 1 is invalid because 1 raised to every real exponent remains 1, so it cannot provide a one-to-one inverse relation.
A base between 0 and 1 is valid and produces a decreasing logarithmic function. Do not reject it merely because classroom examples often use bases greater than 1.
Finite-precision output may underflow or overflow at extreme exponents. The workbench rejects non-finite or non-unique inverse cases rather than displaying a plausible but unsupported result.
Record the chosen base alongside every rounded result and inverse check.

Log Calculator

Solve any part of logᵦ(x) = y

Logarithm relation

Solve one real logarithm relation with every domain condition exposed

Choose which member of the logarithm relation is unknown

Solve for logarithm y

Solve for argument x

Solve for base b

Choose which member of the logarithm relation is unknown

Solve for logarithm y

Solve for argument x

Solve for base b

Verify a base-ten result in exponential form

Distinguish base choice from notation habit

Enforce real-domain exclusions before pressing calculate

Important limitations

Solve any part of logᵦ(x) = y

Logarithm relation

Solve one real logarithm relation with every domain condition exposed

Choose which member of the logarithm relation is unknown

Solve for logarithm y

Solve for argument x

Solve for base b

Choose which member of the logarithm relation is unknown

Solve for logarithm y

Solve for argument x

Solve for base b

Verify a base-ten result in exponential form

Distinguish base choice from notation habit

Enforce real-domain exclusions before pressing calculate