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Logarithms and Exponents Explained

August 2026 · 5 min read

Logarithms and exponents are inverse operations of each other, which is exactly why they're usually taught together — understanding one makes the other much easier to grasp.

What Is an Exponent?

An exponent tells you how many times to multiply a number by itself. In the expression 2³, the base is 2 and the exponent is 3, meaning 2 × 2 × 2 = 8. Exponents grow extremely quickly, which is why they're used to model things like compound interest, population growth, and radioactive decay.

What Is a Logarithm?

A logarithm answers the reverse question: "What power do I need to raise this base to, to get this number?" For example, log₂(8) = 3, because 2 raised to the power of 3 equals 8. In other words, logarithms and exponents undo each other.

Common Log vs Natural Log

The common logarithm (written as log) uses base 10, so log(100) = 2 because 10² = 100. The natural logarithm (written as ln) uses base e (approximately 2.718), and shows up constantly in calculus, growth/decay models, and finance.

Key Logarithm Rules

These rules let you break complicated expressions into simpler pieces before calculating, which is especially useful when solving equations where the unknown is in the exponent.

Real-World Uses

Logarithms measure things that span huge ranges of scale — earthquake intensity (the Richter scale), sound loudness (decibels), and pH in chemistry are all logarithmic scales. This is precisely because logarithms compress very large or very small numbers into a manageable range.

Calculate With Confidence

Calvo's scientific calculator includes both log and ln functions, so you can move between exponential and logarithmic calculations without switching tools.