Logarithm Rules, Maths First, Institute of Fundamental Sciences, Massey University

Rules

Since the notion of a logarithm is derived from exponents, all logarithmic rules for multiplication, division and raised to a power are based on those for exponents. There are no general rules for the logarithms of sums and differences.

Recall that when we

multiply two powers we add their exponents
b^mbⁿ = b^m+n

divide one power by another we subtract the exponents

= b^m−n

raise one power by a number we multiply the exponent by that number.
(b^m)ⁿ = b^mn

Translating these rules to logarithms we obtain:

Rules

For all positive real numbers,

log_b(xy) = log_b x + log_b y

log_b(xⁿ) = n log_b x

with Special Cases:

log_b b = 1, as b¹ = b

log_b 1= 0, as b⁰ = 1

Example

Express the logarithm of a number in terms of the logarithms of as few numbers of primes as possible. Try some of these yourself and check against the given answers. There are 8 examples to view.

log₁₀

decomposition of x

log₁₀

( )

log₁₀x written in terms of logs of primes

log₁₀x written as a decimal

(2 d.p.)

Example

Express the logarithm of an expression in terms of the logarithms of as few numbers of primes as possible. Try some of these exercises yourself and check against the given answers. There are 8 examples given.

log₁₀

decomposition of x

log₁₀

( )

log₁₀x written in terms of logs of primes

log₁₀x written in decimal form

(2 d.p.)

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