Logarithm Calculator: Log Base 10, Natural Log and Any Base

Logarithms are the inverse of exponentials — they answer the question, “To what power must I raise this base to get that number?” They tame enormous ranges of values, which is why they power the pH scale, the Richter scale, decibels, and countless algorithms. A logarithm calculator computes log base 10, natural log, log base 2, or a logarithm of any base in an instant.

This guide explains what logarithms are, how they work, and the laws that make them useful.

What Is a Logarithm?

A logarithm is defined by this relationship:

log_b(x) = y ⟺ bʸ = x

In words, log base b of x is the exponent y that turns b into x. For example:

• log₁₀(1000) = 3, because 10³ = 1000.
• log₂(8) = 3, because 2³ = 8.
• log₅(25) = 2, because 5² = 25.

The Common Bases

Three bases appear most often:

• Base 10 (common log), written log or log₁₀ — used in engineering and the pH/decibel scales.
• Base e (natural log), written ln, where e ≈ 2.71828 — used throughout calculus, growth, and decay.
• Base 2 (binary log) — used in computer science and information theory.

The Change-of-Base Formula

Calculators compute a logarithm of any base using the change-of-base formula, which relies on natural or common logs:

log_b(x) = ln(x) ÷ ln(b) = log(x) ÷ log(b)

This is exactly how the calculator handles arbitrary bases.

A Worked Example

Compute log₃(81):

• Using change of base: log₃(81) = ln(81) ÷ ln(3) ≈ 4.394 ÷ 1.099 ≈ 4.
• Check: 3⁴ = 81. ✓

The Laws of Logarithms

These identities mirror the laws of exponents and are essential for simplifying expressions:

• Product rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient rule: log_b(x ÷ y) = log_b(x) − log_b(y)
• Power rule: log_b(xⁿ) = n · log_b(x)
• Log of 1: log_b(1) = 0 (because b⁰ = 1)
• Log of the base: log_b(b) = 1

How to Use the Logarithm Calculator

1. Enter the value x you want the logarithm of (it must be positive).
2. Enter the base — the default is 10, with quick options for e and 2.
3. The calculator returns the result, for example “log₁₀(1000) = 3.”

Why Logarithms Matter

Logarithms compress vast scales into manageable numbers:

• pH measures acidity as −log₁₀ of hydrogen-ion concentration.
• The Richter scale expresses earthquake energy logarithmically, so each whole step is roughly 10× stronger.
• Decibels measure sound intensity on a log scale.
• Algorithms: binary search and many data structures run in O(log n) time.

Common Mistakes to Avoid

• Taking the log of zero or a negative number. Logarithms are only defined for positive inputs.
• Using base 1. Base 1 is invalid because 1 raised to any power is always 1.
• Confusing ln and log. ln is base e; log usually means base 10 (or base 2 in computer science — always check the context).
• Splitting log of a sum. log(x + y) is not log x + log y.

Conclusion

Logarithms invert exponentials and turn multiplication into addition, which is why they appear across science, engineering, and computing. A logarithm calculator handles any base instantly using the change-of-base formula, so you never have to wrestle with the arithmetic by hand.

Try our free Logarithm Calculator for instant results.