Definition of Logarithms: A Complete Student Guide

The statement log_b a = x is simply another way of writing b^x = a, read as "the logarithm of a to base b is the exponent x". A logarithm answers one question: to what power must b be raised to get a?

Logarithms matter because they turn the hard operation of exponentiation into the easy operations of addition and subtraction — the idea behind the slide rule, and still the reason calculators and computers handle huge ranges of numbers gracefully.

Key Concepts

• Base b: must be positive and not equal to 1; otherwise b^x cannot produce a range of values.
• Argument a: must be positive; log 0 and log of a negative number do not exist as real numbers.
• The logarithm is an exponent: log_2 32 = 5 because the exponent that turns 2 into 32 is 5.
• Two instant values: log_b 1 = 0 since b⁰ = 1, and log_b b = 1 since b¹ = b.
• Inverse pair: b^(log_b a) = a and log_b(b^x) = x — exponentiation and logarithms undo each other.

Formulas

The definition log_b a = x ⇔ b^x = a is the master key: every logarithm problem is solved by translating it into this exponential form. All formulas below require b > 0, b ≠ 1, and a > 0.

• log_b a = x ⇔ b^x = a (the definition)
• log_b 1 = 0 and log_b b = 1
• log_b(b^x) = x
• b^(log_b a) = a

Worked Example

Evaluate log_2 32, log_10 1000, and log_3(1/9) by asking "the base to what power gives the argument?" For log_2 32: 32 = 2⁵, so the answer is 5; check 2⁵ = 32. For log_10 1000: 1000 = 10³, so the answer is 3; check 10³ = 1000. For log_3(1/9): 1/9 = 3^(−2), so the answer is −2; check 3^(−2) = 1/9. Logarithms can be negative or zero — only the argument must stay positive, while the answer (an exponent) is unrestricted.

Common Misconceptions

• log(−5) or log 0 "equals something". No power of a positive base gives zero or a negative number, so neither exists as a real logarithm.
• Any positive number can be a base. Base 1 is forbidden because 1^x = 1 can never produce any other value.
• "log 0 = 0". The zero case is log 1 = 0, since b⁰ = 1 — never log 0.

Real-World Applications

• The pH scale in chemistry: pH = −log[H⁺].
• The Richter scale for earthquake magnitude.
• Decibels for sound intensity.
• Binary logarithms in computer science for halving processes like binary search.

A logarithm is an exponent with a new name: log_b a = x means exactly b^x = a. Keep the base positive and away from 1, keep the argument positive, and every logarithm becomes a small "which power?" puzzle.