Logarithms Explained: They Just Reverse Powers

A logarithm answers a single question: what power do I need to raise this base to, to get this number?

log₁₀(1000) = 3 because 10 to the power of 3 is 1000.

That's the whole idea. Everything else is rules for manipulating that question.

The three rules that matter

• Product rule: log(a × b) = log(a) + log(b)
• Quotient rule: log(a / b) = log(a) - log(b)
• Power rule: log(aⁿ) = n × log(a)

Memorise these three. They cover almost every log question.

When you'll see logs

• Solving exponential equations (when x is in the exponent)
• Acidity (pH is a log scale)
• Sound intensity (decibels)
• Earthquake magnitude (Richter scale)
• Compound interest at long horizons

Worked example

Solve 2ˣ = 50.

Take log of both sides: x × log(2) = log(50). So x = log(50) / log(2) ≈ 5.64.

That's the power rule in action.

Common pitfalls

• log(a + b) ≠ log(a) + log(b). Never. The rule is for multiplication, not addition.
• log(0) is undefined. Negative numbers also have no real log.
• ln means log base e (natural log). It behaves the same way.

If logs still feel slippery, try the math solver on three problems and watch the steps. The pattern clicks fast.