Antilogarithm: from the logarithm back to the number

Undo the logarithm

An antilogarithm is the inverse of a logarithm. If log_b x = y, finding x is called taking the antilogarithm of y.

In 4⁵ = 1024, the base is 4, the logarithm is 5 and the antilogarithm is 1024. The antilogarithm is just the number, or result.

With tables: split the log into characteristic and mantissa. Look up the mantissa in the antilog table and add the mean difference.

Then place the point using the characteristic. Count from the reference position, between the first non-zero digit and the next: right for a positive characteristic, left for a barred one.

A negative log must first be made into a barred characteristic plus a positive mantissa: −1.4567 = 2̄.5433.

Key terms

Antilogarithm

Finding the number from its log.

Antilog table

Turning a mantissa into digits.

Reference position

Where you start counting places for the point.

Barred characteristic

A negative characteristic with a positive mantissa.

Short questions with model answers

1. Q1. Example 11 (ii): find x if log x = −1.4567.

   • −2 + 2 − 1.4567 = −2 + 0.5433 = 2̄.5433
   • table 0.5433 → 3491 + 2 = 3493
   • 2̄ → point 2 places left of 3|493 → 0.03493

   x = 0.03493.

2. Q2. Example 11 (iii): find x if log x = −2.1234.

   • −3 + 3 − 2.1234 = −3 + 0.8766 = 3̄.8766
   • table 0.8766 → 7516 + 10 = 7526
   • 3̄ → 3 places left of 7|526 → 0.007526

   x = 0.007526.

Common mistakes

• ✗ Looking up 0.4567 in the table for log x = −1.4567.

  ✓ First rewrite as 2̄.5433; look up 0.5433.

• ✗ Counting places from the start of 1332: 13.32.

  ✓ Count from the reference position after the 1: 133.2.

• ✗ Using the log table to find an antilog.

  ✓ Antilogs need the antilog table, with rows .00 to .99.

MCQs

1. 1. In 4³ = 64, the antilogarithm is:

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

   (d) The result is the antilogarithm (p. 28).

2. 2. antilog 2.1245 =

   1. (a) 13.32
   2. (b) 133.2
   3. (c) 1332
   4. (d) 1.332
   Show answer

   (b) Characteristic 2: two places right of the 1 (p. 28).

3. 3. −1.4567 written with a positive mantissa is:

   1. (a) 2̄.5433
   2. (b) 1̄.4567
   3. (c) 1̄.5433
   4. (d) 2̄.4567
   Show answer

   (a) −2 + 0.5433 (Example 11 ii).

4. 4. If log x = 0.2568, then x is:

   1. (a) 0.1806
   2. (b) 18.06
   3. (c) 1.806
   4. (d) 180.6
   Show answer

   (c) Characteristic 0: point right at the reference position (Example 11 i).

5. 5. If log x = 3̄.8766, then x is:

   1. (a) 0.07526
   2. (b) 0.007526
   3. (c) 7526
   4. (d) 0.0007526
   Show answer

   (b) Three places left of the reference position (Example 11 iii).

Quick revision

• The antilogarithm turns a logarithm back into the number.
• Mantissa → antilog table + mean difference → digits.
• The characteristic places the point from the reference position.