# Three Laws of Logarithms

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Many standardized tests will challenge you with these three laws of logarithms.

Law 1: The logarithm of a product is equal to the sum of the logarithms of each term in the product.

log b xy = log b x + log b y

• ﻿log2 3(4) = ;log2 3 + log2 4
• log7 5(6) = ;log7 5 + log7 6
• log15 8(12) = ;log15 8 + log15 12
• log11 9(7) = ;log11 9 + log11 7

Law 2: The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.

log b x/y = log b x – log b y

• ﻿log2 5/6 = ;log2 5 - log2 6
• log3 7/8 = ;log3 7 - log3 8
• log5 3/4 = ;log5 3 - log5 4
• log7 6/11 = ;log7 6 - log7 11

Law 3: The logarithm of x with a rational exponent is equal to the exponent times the logarithm.

log b xn = n log b x

• ﻿log2 56 = ;6 log2 5
• log3 78 = ;8 log3 7
• log5 123 = ;3 log5 12
• log9 97 = ;7 log9 9
• log3 71/4 = ;1/4 log3 7
• log9 97/3 = ;7/3 log9 9
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