log calculator

What is a logarithm?

A logarithm tells you the exponent needed to raise a base to get a number. If b^y = x, then log_b(x) = y.

Example: because 10³ = 1000, we know log₁₀(1000) = 3. Logarithms are the inverse of exponentiation, and they show up everywhere: in science, engineering, finance, statistics, and computer science.

How to use this log calculator

Step-by-step

• Select a log type: base 10, natural log (base e), base 2, or custom base.
• Enter the number x you want the logarithm of.
• Confirm the base b (for custom base, choose any valid value).
• Set your preferred decimal precision.
• Click Calculate to get the result instantly.

Important input rules

• x must be greater than 0
• b must be greater than 0
• b cannot be 1 (because log base 1 is undefined)

Common logarithm types

Common log: log₁₀(x)

The common log uses base 10 and is widely used in measurement scales and quick order-of-magnitude comparisons.

Natural log: ln(x)

The natural logarithm uses base e (approximately 2.718281828). It appears in growth and decay models, compound processes, and calculus.

Binary log: log₂(x)

Base-2 logarithms are common in computer science, where doubling and halving patterns are fundamental.

Worked examples

Example 1: log₁₀(1000)

Because 10³ = 1000, the answer is 3.

Example 2: ln(20)

ln(20) is approximately 2.995732, meaning e^2.995732 is about 20.

Example 3: log₂(64)

Since 2⁶ = 64, log₂(64) = 6.

Where logs are useful in real life

• Finance: continuous compounding and growth modeling
• Data science: log transforms to reduce skewness
• Biology: population growth and decay
• Engineering: signal and scale analysis
• Computer science: algorithm complexity like O(log n)

Quick tips for accuracy

• Use more decimal places for sensitive calculations.
• Double-check the base before interpreting the output.
• Remember that changing the base changes the value.
• Verify with inverse exponentiation if needed.

Final thoughts

Logarithms look intimidating at first, but they are just exponent questions in reverse. A reliable log calculator helps you move from theory to action quickly—whether you are solving homework problems, validating data pipelines, or analyzing growth trends.