Logarithm: the power you need, log_b x = y ⇔ bʸ = x

A logarithm is an exponent

Your book says the word comes from the Greek logos and arithmos, meaning ratio or proportion. The Scottish mathematician John Napier introduced it to simplify multiplying and dividing large numbers.

The logarithm of x to the base b is y, written log_b x = y. It means b raised to the power y equals x: bʸ = x.

In that statement b is the base, x is the result and y is the exponent. The base must be positive and not 1, and x must be positive.

Every log statement can be rewritten as a power, and back. That switch solves most of the questions in this section.

Read log_b x as: what power of b gives x? The answer to that question is the logarithm.

Key terms

Base (b)

The number being raised to a power.

Exponent (y)

The power that gives the result.

Exponential form

The same fact written as a power.

Logarithmic form

The same fact asking for the power.

Short questions with model answers

1. Q1. Example 7 (i): find x if log₅25 = x.

   • 5ˣ = 25
   • 5ˣ = 5² → x = 2

   x = 2.

2. Q2. Example 8: write 3⁴ = 81 and 7⁰ = 1 in logarithmic form.

   • 3⁴ = 81 → log₃81 = 4
   • 7⁰ = 1 → log₇1 = 0

   log₃81 = 4 and log₇1 = 0.

Common mistakes

• ✗ Writing 3⁴ = 81 as log₄81 = 3.

  ✓ The base of the power is the base of the log: log₃81 = 4.

• ✗ Reading log₂x = 6 as x = 2 × 6 = 12.

  ✓ A log gives a power, not a product: x = 2⁶ = 64.

• ✗ Writing log 0 = 0.

  ✓ x must be positive; no power of b gives 0, so log 0 is undefined.

MCQs

1. 1. 3⁴ = 81 in logarithmic form is:

   1. (a) log₃4 = 81
   2. (b) log₄3 = 81
   3. (c) log₃81 = 4
   4. (d) log₄81 = 3
   Show answer

   (c) Base 3, power 4 (Review Exercise 2 x).

2. 2. log₂8 = 3 in exponential form is:

   1. (a) 2³ = 8
   2. (b) 3² = 8
   3. (c) 8³ = 2
   4. (d) 2⁸ = 3
   Show answer

   (a) Example 5, p. 23.

3. 3. If log₂x = 6, then x =

   1. (a) 12
   2. (b) 36
   3. (c) 3
   4. (d) 64
   Show answer

   (d) x = 2⁶ = 64 (Example 7 ii).

4. 4. log₂2³ equals:

   1. (a) 2
   2. (b) 3
   3. (c) 8
   4. (d) 6
   Show answer

   (b) 2 to the power 3 gives 2³, so the log is 3 (Review Exercise 2 iv).

5. 5. Which condition does your book put on the base b?

   1. (a) b > 0 and b ≠ 1
   2. (b) b < 0
   3. (c) b = 1
   4. (d) b = 0
   Show answer

   (a) b > 0, x > 0 and b ≠ 1 (p. 23).

Quick revision

• log_b x = y means bʸ = x, with b > 0, b ≠ 1 and x > 0.
• The base stays the base; the power becomes the logarithm.
• Switching between the two forms solves for an unknown.