Logarithm Calculator

Calculate logarithms in any base including log base 10, natural log (ln), and custom bases. See step-by-step solutions using the change-of-base formula.

What is Logarithm Calculator?

The logarithm is one of the most powerful and versatile mathematical concepts, forming the foundation of everything from earthquake measurement to signal processing, from information theory to chemical pH. At its core, log_b(n) asks: "What power must I raise b to in order to get n?" When b = 10, this is the common logarithm; when b = e ≈ 2.718, it is the natural logarithm — the language of growth, decay, and continuous change.

This calculator handles all three standard logarithm types — log base 10, natural log (ln), and any custom base — and displays the full step-by-step solution using the change-of-base formula. Understanding this formula is key: every logarithm can be expressed as a ratio of two natural logs, making computation possible with any scientific calculator.

Formula

Definition: log_b(n) = x ↔ b^x = n

Change of base: log_b(n) = ln(n) / ln(b) = log₁₀(n) / log₁₀(b)

Logarithm rules:

Product: log_b(xy) = log_b(x) + log_b(y)

Quotient: log_b(x/y) = log_b(x) − log_b(y)

Power: log_b(xⁿ) = n × log_b(x)

Special: log_b(1) = 0, log_b(b) = 1, ln(e) = 1

How to Calculate

Enter the number n (must be positive — logarithm of zero or negatives is undefined).
Select your base: log₁₀ for common logarithm, ln for natural logarithm, or enter a custom base.
The calculator applies the change-of-base formula: log_b(n) = ln(n) / ln(b).
Verify the result by checking: b^result ≈ n (shown as antilog).
Use the reference table to compare logarithm values across different bases.

Example

Find log₅(125): log₅(125) = ln(125)/ln(5) = 4.8283/1.6094 = 3. Verification: 5³ = 125 ✓. This makes intuitive sense because 125 = 5³.

Key Terms Explained

Logarithm: The exponent to which a base must be raised to produce a given number
Common logarithm (log₁₀): Base-10 logarithm, used in pH and decibels
Natural logarithm (ln): Base-e logarithm, fundamental in calculus and growth models
Antilogarithm: The inverse of a logarithm — b raised to the power of the log result
Change-of-base formula: log_b(n) = ln(n)/ln(b) — converts any log to computable form
Euler's number (e): Mathematical constant ≈ 2.71828, base of the natural logarithm

Common Use Cases

Solving exponential equations in algebra and calculus
Computing pH of a solution: pH = −log₁₀[H⁺]
Calculating decibel levels: dB = 10×log₁₀(P₁/P₀)
Analyzing Richter scale earthquake magnitudes
Computing information entropy: H = −Σ p×log₂(p)
Continuous compound interest: A = Pe^(rt)

Frequently Asked Questions

What is a logarithm?

A logarithm answers: "To what exponent must I raise the base to get this number?" If log_b(n) = x, then b^x = n. For example, log₁₀(1000) = 3 because 10³ = 1000.

What is the natural logarithm (ln)?

The natural logarithm (ln) uses Euler's number e ≈ 2.71828 as its base. It appears in calculus, compound interest, population growth, radioactive decay, and physics. ln(x) = log_e(x).

What is the change-of-base formula?

The change-of-base formula: log_b(n) = ln(n)/ln(b) = log₁₀(n)/log₁₀(b). This means any logarithm can be computed using only ln or log₁₀.

Why are logarithms used in science?

Logarithms compress huge ranges into manageable scales. The Richter scale, decibel scale, pH scale, and stellar magnitude all use logarithms. In computing, O(log n) algorithms are among the most efficient possible.

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