Logarithms — Mathematics, 14–17 years
The reverse operation to powers: finding which exponent produces a given number. Mathematics, 14–17 years.
The idea
A logarithm asks, “Which exponent do I need?” The statement log₂ 32 = 5 means 2^5 = 32. Logs turn multiplication of exponents into a question we can solve, especially when the unknown is in the exponent rather than beside it.
Why they were needed
Before calculators, long multiplications were slow and easy to get wrong. Logarithm tables let people replace multiplication by addition, making astronomy, navigation and engineering more manageable. Today calculators do the arithmetic, but logs still reveal scales and solve exponential equations.
Solving one
Solve 3^x = 81. Ask which power of 3 gives 81: 3^1 = 3, 3^2 = 9, 3^3 = 27 and 3^4 = 81. Therefore x = log₃ 81 = 4. The logarithm names the exponent we found by testing powers.
The common trap
It is easy to read log₂ 8 as 2 × 8, because the small number looks like a multiplier. But the base 2 is the repeated factor, and the answer is an exponent: 2^3 = 8, so log₂ 8 = 3. Switching the base and the answer changes the question completely.
Where they appear
Logarithmic scales are useful when values cover a huge range. Earthquake magnitude, sound level in decibels and acidity measured by pH all compare powers rather than ordinary differences. A one-unit rise on such a scale does not usually mean “one more”; it represents a fixed multiplication factor.
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