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Common logarithms

The common logarithm lg⁡x\lg x answers which exponent 10 must have to give xx. So lg⁡\lg and the power 10y10^y are inverse operations. The logarithm turns a product into a sum and a power into a product. There is no corresponding rule for lg⁡(a+b)\lg(a + b). The argument must be positive: the logarithm is not defined for zero or negative numbers.

lg⁡x=y  ⟺  10y=x\lg x = y \iff 10^y = xthe definition
lg⁡(ab)=lg⁡a+lg⁡b\lg(ab) = \lg a + \lg ba product becomes a sum
lg⁡an=nlg⁡a\lg a^n = n\lg aa power becomes a product

Symbols

xxa positive number
yythe exponent

Example

lg⁡1000=3\lg 1000 = 3 and lg⁡0.01=−2\lg 0.01 = -2. lg⁡2+lg⁡5=lg⁡10=1\lg 2 + \lg 5 = \lg 10 = 1.

Rewrite the number as a power of 10 before you read off the logarithm.
Practise powers, roots and logarithms for free →

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Part of Mathematics 1T: Powers, roots and logarithms.