Log Calculator

What is a log calculator?

A log calculator finds the exponent that a base must be raised to in order to produce a target value. It runs the standard logarithm in any base, the common log (base 10), the natural log (base e), and works in reverse to solve for the missing exponent in an exponential equation.

How the log calculator works

Pick a mode and enter the inputs:

• Log with custom base: enter a value and a base. The calculator returns log_b(x) using the change-of-base formula log_b(x) = ln(x) / ln(b).
• Common log: enter a value. The calculator returns log₁₀(x).
• Natural log: enter a value. The calculator returns ln(x) = log_e(x).
• Solve missing exponent: enter a base and a result. The calculator returns the exponent y such that b^y = result, using the same change-of-base formula.

Every mode also shows the equivalent exponent statement and a plain English explanation, so you can see the relationship from both sides.

Log formula

The defining relationship is log_b(x) = y if and only if b^y = x. The change-of-base formula expresses any logarithm in terms of natural log or common log:

• log_b(x) = ln(x) / ln(b)
• log_b(x) = log₁₀(x) / log₁₀(b)

A handful of identities make hand calculation easier:

• log_b(1) = 0 for any base
• log_b(b) = 1 for any base
• log_b(x · y) = log_b(x) + log_b(y)
• log_b(x / y) = log_b(x) − log_b(y)
• log_b(x^k) = k · log_b(x)

Logs vs exponents

Logarithms and exponents are inverse operations. An exponent raises a base to a power by repeated multiplication. A logarithm asks the opposite question: given a base and a result, what exponent produces the result?

• Exponent: 2^3 = 8 (base 2, exponent 3, result 8)
• Logarithm: log₂(8) = 3 (base 2, result 8, exponent 3)

Both statements describe the same relationship from different directions. Use the exponent calculator when the base and exponent are known and you want the result. Use this log calculator when the base and result are known and you want the exponent. The Solve missing exponent mode in this calculator is exactly the logarithm framed as the inverse of exponentiation.

Common log vs natural log

The common log uses base 10 and is the standard tool in chemistry (pH), audio (decibels), and earthquake magnitude (Richter scale). The natural log uses base e (about 2.71828) and is the standard tool in calculus, continuous growth and decay, and many physics formulas. Both are special cases of the general logarithm; the calculator exposes each as its own mode for convenience.

When a math text writes plain log without a base, it usually means log₁₀ in introductory and engineering settings, or ln in some pure-math settings. The calculator labels each mode explicitly so there is no ambiguity in the result.

Worked examples

• log₂(8) = 3, because 2 × 2 × 2 = 8.
• log₁₀(1000) = 3, because 10 × 10 × 10 = 1000.
• log₁₀(0.01) = −2, because 10^(−2) = 1 / 100 = 0.01.
• ln(e) = 1, because e^1 = e.
• ln(1) = 0 for any base.
• log₂(16) = 4, the answer to 2^y = 16.

Common mistakes

• Taking a log of zero or a negative number. Logarithms are only defined for positive real values. The calculator flags these cases.
• Using a base of 1. Every power of 1 is 1, so the logarithm with base 1 has no unique answer; the base must be greater than 0 and not equal to 1.
• Confusing log with ln. In most introductory and engineering contexts, log means base 10, and ln means base e. The calculator labels each mode so the base is always explicit.
• Treating log_b(x + y) as log_b(x) + log_b(y). The log of a sum is not the sum of logs. The product rule applies to products: log_b(x · y) = log_b(x) + log_b(y).
• Forgetting that log_b(x^k) = k · log_b(x) pulls the exponent out front. This is the identity most useful for solving exponential equations.

Where logarithms show up

Logarithms appear any time you need to solve for an exponent, compress a wide range of values onto a readable scale, or model exponential change. Common applications include pH and decibel scales, the Richter scale, half-life and continuous-growth formulas, information theory (bits use log base 2), and algorithm analysis (O(log n) complexity). They also show up in the compound-interest formula when you solve for time.

Related tools

• Scientific calculator for a full button-driven calculator with log, ln, sin, cos, tan, and exponents in one place.
• Exponent calculator runs the inverse operation: base raised to a power.
• Square root calculator for the special case of the 1/2 power.
• Factor calculator for prime factorization, factor pairs, and the greatest common factor.
• All education calculators.