Logarithm Rules

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Logarithm Rules: meaning, method and solved example (Algebra 2) – Maths PDF

In short: The logarithm rules let a log swallow a product as a sum, a fraction as a difference and a power as a plain multiplier, so you can expand, condense and estimate logs by hand.

Algebra 2Functions and GraphsF-LE.A.4

What is Logarithm Rules?

The logarithm rules copy the laws of exponents into log language. A log is an exponent, so when numbers are multiplied their exponents get added, when they are divided the exponents get subtracted, and a power scales the exponent.

For a base \(b > 0\), \(b \neq 1\), and positive \(M\) and \(N\): the product rule is \(\log_b(MN) = \log_b M + \log_b N\); the quotient rule is \(\log_b\frac{M}{N} = \log_b M – \log_b N\); the power rule is \(\log_b(M^p) = p\log_b M\). Also \(\log_b 1 = 0\) and \(\log_b b = 1\).

How to spot it

  • One log hides a product, a quotient or a power inside, and the directions say to expand it with the logarithm rules.
  • Several logs with the same base are added or subtracted and the directions say condense.
  • You know the values of a couple of logs and must estimate another one without a calculator.
  • An unknown sits in an exponent and must be brought down so it can be solved.

The method

  1. Confirm that all the logs share the same base and that every inside quantity is positive.
  2. To expand, split products with addition, quotients with subtraction, and move powers in front.
  3. To condense, move coefficients up as powers, then merge sums into one product and differences into one quotient.
  4. Evaluate any piece that is an exact power of the base, such as \(\log_2 32 = 5\).
product: multiply inside, add outsidelog(M x N) = log M + log Nquotient: divide inside, subtract outsidelog(M / N) = log M – log Npower: exponent inside, multiplier outsidelog(M^p) = p x log M
Three rules, one idea: logs are exponents, so they follow the laws of exponents.

Logarithm Rules: solved example

Problem

Given \(\log_5 2 \approx 0.4307\) and \(\log_5 3 \approx 0.6826\), estimate \(\log_5 12\) and \(\log_5 \frac{2}{3}\).

Solution

Split 12 as \(2 \cdot 2 \cdot 3\): \(\log_5 12 = 2\log_5 2 + \log_5 3 \approx 2(0.4307) + 0.6826 = 1.5440\).

For the fraction, subtract: \(\log_5 \frac{2}{3} = \log_5 2 – \log_5 3 \approx 0.4307 – 0.6826 = -0.2519\). The estimates are 1.5440 and \(-0.2519\).

Try it

Question 1

Condense \(\log_4 6 + \log_4 8 – \log_4 3\) into one number.

Show the answer

\(\log_4 \frac{6 \cdot 8}{3} = \log_4 16 = 2\).

Question 2

Expand \(\log_7 (49m^3)\).

Show the answer

\(\log_7 49 + 3\log_7 m = 2 + 3\log_7 m\).

Logarithm Rules vs. Logarithm

A logarithm is one quantity: the exponent that a base needs. These rules are the laws governing how several such quantities combine. A common slip is writing \(\log(M + N) = \log M + \log N\); no rule covers a sum inside the log, and the true identity applies only to products.

Logarithm

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