Logarithm Calculator

What the Logarithm Result Means

A logarithm answers an exponent question. If log_b(x) = y, then b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000.

The answer can be negative or fractional. A negative logarithm is valid when the input lies between 0 and 1 for a base greater than 1. For example, log₁₀(0.01) = -2.

Logarithm Formula

For a positive number x and a valid base b, the logarithm is written as:

Change-of-base formula

A logarithm with almost any valid base can be calculated using natural logarithms:

The same relationship also works with common logarithms: log_b(x) = log₁₀(x) / log₁₀(b).

Valid inputs

• x must be greater than 0. Real-valued logarithms are not defined for zero or negative inputs.
• The base must be greater than 0.
• The base cannot equal 1. Powers of 1 never produce values other than 1, so they cannot define a usable logarithm function.
• A base between 0 and 1 is allowed. For example, log_0.5(0.25) = 2.

Common, Natural, and Binary Logarithms

How to Use the Calculator

1. Enter the positive number whose logarithm you want to find.
2. Select base 10, base e, base 2, or Custom base.
3. If you choose a custom base, enter a positive number other than 1.
4. Select how many decimal places may be shown.
5. Press Calculate Logarithm.
6. Read the logarithm, equivalent exponential form, and calculation steps in the result box.

Worked Example: log₂(32)

Suppose you want to calculate log₂(32). This asks: “What exponent on 2 produces 32?”

1. ln(32) ≈ 3.465736
2. ln(2) ≈ 0.693147
3. 3.465736 / 0.693147 = 5

Therefore, log₂(32) = 5. The exponential check is 2⁵ = 32.

Where Logarithms Are Used

Logarithms are useful whenever values grow or shrink by repeated multiplication rather than simple addition.

• Algebra: solving equations in which the unknown appears in an exponent.
• Science: working with exponential growth, decay, and measurements expressed on logarithmic scales.
• Computing: base-2 logarithms help describe powers of two and repeated halving or doubling.
• Data analysis: logarithmic transformations can make very wide numerical ranges easier to compare.
• Compound growth: logarithms can isolate an unknown time or exponent in exponential formulas.

Logarithm Rules That Help With Manual Calculations

For positive values where the expressions are defined, these identities can simplify logarithmic expressions:

Two useful reference values are log_b(1) = 0 and log_b(b) = 1.

Common Input Mistakes

• Entering zero: a real logarithm of zero does not exist.
• Entering a negative number: this calculator works with real-valued logarithms, so the input must be positive.
• Using base 1: 1 is not a valid logarithm base.
• Confusing ln with log base 10: ln uses base e, while the common logarithm uses base 10.
• Rounding too early: keeping more digits during intermediate calculations produces a more accurate final result.

Precision and Calculator Limits

This calculator uses JavaScript floating-point arithmetic. Most everyday logarithm calculations are displayed accurately to the selected number of decimal places, but values can contain tiny rounding differences. An exact mathematical result such as 3 may internally be represented by a nearby decimal before formatting.

Inputs must also fit within the numeric range supported by the browser. The tool rejects blank entries, non-finite values, invalid bases, zero, and negative inputs.

Frequently Asked Questions