Laws of logarithm: product, quotient, power and change of base

Multiplying becomes adding

Your book gives four laws that simplify log expressions and solve log equations. Each one comes straight from a law of indices.

Product law: log_b xy = log_b x + log_b y. Quotient law: log_b (x/y) = log_b x − log_b y. The log of a product is a sum; of a quotient, a difference.

Power law: log_b xⁿ = n log_b x, so a power comes down in front. Change of base: log_b x = log_a x / log_a b, which lets you use any base.

The proofs all start the same way: let m = log_b x and n = log_b y, so x = bᵐ and y = bⁿ. Then xy = b^(m+n) gives the product law.

To expand: break products and quotients apart, then bring powers down. To combine: lift numbers in front up as powers first, then join.

Key terms

Product law

Splitting a product into a sum.

Quotient law

Splitting a quotient into a difference.

Power law

Bringing a power down in front.

Change of base

Rewriting a log in a base you can use.

Short questions with model answers

1. Q1. Example 13 (ii): expand log₅(xy/z)⁸.

   • 8 log₅(xy/z)
   • 8[log₅(xy) − log₅z]
   • 8[log₅x + log₅y − log₅z]

   8[log₅x + log₅y − log₅z].

2. Q2. Example 15: L = 40 log₁₀(I/I₀) as printed in your book. Find L when I = 10⁶ I₀.

   • L = 40 log₁₀(10⁶ I₀ / I₀) = 40 log₁₀10⁶
   • L = 40 × 6 log₁₀10 = 40 × 6 = 240

   240 decibels, as your book says. Teacher note: the usual formula has 10, not 40, which gives 60 dB.

Common mistakes

• ✗ log 5 + log 3 = log 8.

  ✓ A sum of logs is the log of a product: log 15.

• ✗ 2 log₃10 − log₃4 = log₃(10/4) × 2.

  ✓ First lift the 2 up as a power: log₃10² = log₃100, then divide: log₃25.

• ✗ log x / log y = log x − log y.

  ✓ The quotient law is about log (x/y), not a fraction of two logs; log x / log y is change of base.

MCQs

1. 1. log 5 + log 3 =

   1. (a) log 15
   2. (b) log 8
   3. (c) log 2
   4. (d) log (5/3)
   Show answer

   (a) Product law (Review Exercise 2 ix).

2. 2. log₂9⁵ expands to:

   1. (a) 5 log₂3
   2. (b) 45 log₂3
   3. (c) 10 log₂3
   4. (d) log₂45
   Show answer

   (c) 9⁵ = 3¹⁰ (Example 12 ii).

3. 3. 6 log₃x + 2 log₃11 as a single log is:

   1. (a) log₃(22x⁶)
   2. (b) log₃(66x)
   3. (c) log₃(x⁶ + 121)
   4. (d) log₃(121x⁶)
   Show answer

   (d) Powers up, then product law (Example 14 ii).

4. 4. M = log₁₀(A/A₀) with A = 10,000 and A₀ = 10 gives M =

   1. (a) 4
   2. (b) 3
   3. (c) 1000
   4. (d) 1
   Show answer

   (b) 10,000/10 = 1000 = 10³ (Exercise 2.4 Q6).

5. 5. log₃₂27 by change of base is:

   1. (a) (5/3) log₂3
   2. (b) log 27 − log 32
   3. (c) (3/5) log₂3
   4. (d) 27/32
   Show answer

   (c) log 3³ / log 2⁵ = (3/5) log₂3 (Example 12 iii).

Quick revision

• log xy = log x + log y and log (x/y) = log x − log y.
• log xⁿ = n log x, and log_b x = log_a x / log_a b.
• To combine logs, lift the numbers in front up as powers first.