Logarithm Calculator – Natural Log, Common Log & Any Base

A clear answer before the details

The logarithm log_b(x) is the exponent y that makes bʸ = x. The base must be positive and not equal to 1, and x must be positive.

A log calculator with base and visible working

A logarithm calculator answers the inverse question to exponentiation: which power of b produces x? For log₂(8), the answer is 3 because 2³ = 8. Use the any-base mode for a log calculator with base, common log for base 10, or natural logarithm calculator mode for base e. The change-of-base rule log_b(x) = ln(x) ÷ ln(b) explains how the decimal result is produced.

Natural logarithms and valid inputs

The natural logarithm, written ln(x), uses the constant e ≈ 2.71828 and is common in continuous growth, decay, calculus, finance, and science. The argument x must be positive. A real logarithm base must also be positive and not equal to 1, so zero and negative bases are rejected. Use the logarithm calculator with steps to connect the result back to the original exponential equation.

Formula guide

The key equations are separated below. Start with the calculator mode, then choose the shape or model that matches your problem.

Any base

log_b(x) = ln(x) ÷ ln(b)

The base b is positive and cannot equal 1.

Natural logarithm

ln(x) = logₑ(x)

The natural-log base is e ≈ 2.71828.

Common logarithm

log(x) = log₁₀(x)

Common log uses base 10.

Worked example

log₂(8) = 3 because 2³ = 8. Using change of base gives ln(8) ÷ ln(2) = 3.

Formulas for common use cases

Use the matching equation for the problem you are solving, then enter its variables in the calculator above.

Exponential equations

bʸ = x ⇔ y = log_b(x)

A logarithm finds the missing exponent.

Continuous growth

A(t) = A₀eᵏᵗ

Natural logs isolate time or a growth constant.

Sound and acidity scales

pH = −log₁₀[H⁺]

Logarithmic scales compress very large ranges.

How to use this calculator

Choose Any base, Common log, or Natural log (ln).
Enter a positive argument and a valid base when needed.
Read the exact relationship and decimal result.
Use the natural-log shortcut or inverse relationship to check an exponent.

Where it helps

Logarithms help compare exponential growth, solve equations, measure sound or acidity, and understand algorithms. The calculator keeps the notation plain so students can see why a base-2 logarithm of 8 is 3 instead of treating the answer as a black box.

Common mistakes

A base of 0, a negative base, or base 1 is invalid for this real-number calculator. The argument cannot be zero or negative. Do not confuse ln with log10; both are valid but have different bases.

Frequently asked questions

What is a logarithm?

A logarithm answers the question: which exponent on a base produces this positive argument?

What bases are allowed?

The base must be positive and cannot equal 1. The argument must also be positive.

What is ln?

ln is the natural logarithm with base e, approximately 2.71828.

How do I calculate a natural logarithm?

Choose natural log, enter a positive x, and read ln(x). Natural logarithms use base e, where e is approximately 2.71828.

Why can’t I take the logarithm of zero?

There is no real exponent that makes a positive logarithm base equal to zero. As x approaches zero from the right, ln(x) decreases without bound, so x must be positive.

What is log base 10?

The common logarithm log₁₀(x) uses base 10. It asks which power of 10 produces x, and is the default “log” convention in many school and scientific contexts.

Can I share a calculation?

Yes. Use Share to copy a URL containing the current calculator values, then open that link on another device or send it to someone else.