logarithms calculator

What Is a Logarithm?

A logarithm is the inverse of exponentiation. If you know that b^y = x, then y = log_b(x). In plain language, a logarithm answers the question:

“To what power do I raise the base to get this number?”

For example, because 10³ = 1000, we can say log₁₀(1000) = 3.

How to Use This Calculator

Mode 1: Find logarithm

• Choose Find logarithm.
• Enter the number x (must be greater than 0).
• Enter the base b (must be greater than 0 and not equal to 1).
• Click Calculate to get y = log_b(x).

Mode 2: Find antilog

• Choose Find antilog.
• Enter the exponent y.
• Enter the base b.
• Click Calculate to get x = b^y.

Important Logarithm Rules

These rules are useful in algebra, calculus, statistics, and data science:

• Product rule: log_b(MN) = log_b(M) + log_b(N)
• Quotient rule: log_b(M/N) = log_b(M) - log_b(N)
• Power rule: log_b(M^k) = k · log_b(M)
• Change of base: log_b(x) = log(x) / log(b)
• Special values: log_b(1) = 0 and log_b(b) = 1

Common Bases and Their Uses

Base 10 (Common Log)

Used in scientific notation, pH calculations, and engineering scales such as decibels.

Base e (Natural Log, ln)

The natural logarithm uses Euler’s number (approximately 2.718281828). It appears in growth and decay models, compound interest, calculus, and differential equations.

Base 2 (Binary Log)

Common in computer science, algorithm analysis, and information theory.

Worked Examples

Example 1: log₂(32)

We ask: 2 raised to what power equals 32? Since 2⁵ = 32, the answer is 5.

Example 2: log₁₀(0.01)

Because 10^-2 = 0.01, the logarithm is -2.

Example 3: antilog with base 3 and exponent 4

Compute 3⁴ = 81. So the antilog result is 81.

Where Logarithms Show Up in Real Life

• Finance: continuous compounding and growth modeling
• Biology: population growth and reaction kinetics
• Chemistry: pH scale is logarithmic
• Audio: decibel measurements use logs
• Earth science: earthquake magnitude scales are logarithmic
• Machine learning: loss functions and probability transforms often use logs

Common Mistakes to Avoid

• Trying to compute a log of a non-positive number (x ≤ 0).
• Using an invalid base (b ≤ 0 or b = 1).
• Mixing up log and antilog operations.
• Forgetting that logarithm outputs can be negative or fractional.

Quick Summary

Logarithms convert multiplication-style growth into simpler addition-style math. They are powerful for analyzing scale, growth, and complexity. Use the calculator above to switch between logarithm and antilogarithm instantly, verify homework problems, or build intuition for how exponents and logs are connected.