This lesson is all about using logarithm tables to solve complex calculations involving decimals, powers, and roots. Mastering the special rules for decimals (numbers between 0 and 1) is the key to success here.

1. Logarithms for Decimals (Numbers between 0 and 1)

When a number is less than 1, its logarithm has a negative characteristic. We use a special "bar" notation to show this.

• The Rule: The characteristic of log(x) where 0 < x < 1 is found by counting the number of zeros immediately after the decimal point. Add 1 to this count and put a bar over it.

  • 0.5432 -> 0 zeros -> Characteristic is

  • 0.05432 -> 1 zero -> Characteristic is 2

  • 0.005432 -> 2 zeros -> Characteristic is 3

• Important: The mantissa (the decimal part from the log table) is always positive.

• What it means: 3.7350 is not -3.7350. It actually means -3 + 0.7350.

2. Finding the Antilog (with Bar Notation)

The negative characteristic tells you how many places to move the decimal point to the left.

• Method:

  1. Find the number corresponding to the mantissa from the antilog table (e.g., for .7421 the number is 5.522).

  2. The bar number tells you the power of 10. 2 means 10−2.

  3. Combine them: 5.522 x 10⁻² = 0.05522.

• Quick Tip: antilog of 2.xxxx will have one zero after the decimal point. antilog of 3.xxxx will have two zeros.

3. Arithmetic with Logarithms (The Most Important Skill)

You must handle the negative characteristic (bar part) and positive mantissa (decimal part) separately.

a) Addition

Add characteristics and mantissas separately. If the mantissa sum is more than 1, carry the whole number over.

• Example: 3.8753+1.3475

  • Mantissas: 0.8753+0.3475=1.2228

  • Characteristics: 3+1=−3+1=−2

  • Combine: −2+1.2228=−2+1+0.2228=−1+0.2228=1.2228

b) Subtraction

Subtract mantissas. If you need to "borrow", you take 1 from the characteristic and add 1 to the mantissa.

• Example: 0.5143−1.9143

  • Cannot do 0.5143−0.9143. So, borrow from the 0 characteristic.

  • Rewrite 0.5143 as \overline{1} + 1.5143.

  • Characteristics: 1−1=−1−(−1)=0.

  • Mantissas: 1.5143−0.9143=0.6000.

  • Answer: 0.6000

c) Multiplication by an Integer

Multiply the characteristic and mantissa separately, then combine.

• Example: 2.7512×3

  • Characteristic: 2×3=−2×3=−6.

  • Mantissa: 0.7512×3=2.2536.

  • Combine: −6+2.2536=−6+2+0.2536=−4+0.2536=4.2536.

d) Division by an Integer (The Tricky Part!)

You must adjust the characteristic to be perfectly divisible by the number.

• Method: To divide a.bcde by n, change a to the next lowest multiple of n. Balance this by adding to the mantissa.

• Example: Solve 1.5412÷2

  1. -1 is not divisible by 2.

  2. The next lowest multiple of 2 is -2.

  3. Rewrite \overline{1} as \overline{2} + 1.

  4. The problem is now: (2+1.5412)÷2.

  5. Divide each part: 2÷2=1. And 1.5412÷2=0.7706.

  6. Combine the results: 1.7706.

4. Putting It All Together: A Full Example

Problem: Simplify 0.875​0.9872​

1. Set up the log equation: Let P=0.875​0.9872​. logP=2log(0.987)−21​log(0.875)

2. Find values from the table:

   • log(0.987) = \overline{1}.9943

   • log(0.875) = \overline{1}.9420

3. Substitute and calculate:

   • 2×1.9943=2+1.9886=1.9886

   • 21​×1.9420=(2+1.9420)÷2=1.9710

   • logP=1.9886−1.9710

   • logP=(−1+0.9886)−(−1+0.9710)=0+0.0176=0.0176

4. Find the antilog:

   • P=antilog(0.0176)=1.041

Exam Tips & Common Mistakes

• Mistake 1: Division! The division of logarithms with bar notation is the most common place for errors. Always use the method of adjusting the characteristic. Never just divide the number as if it were a simple decimal.

• Mistake 2: Forgetting to Subtract Mean Differences for Cosine. In the tables, mean differences are added for Sine and Tangent, but SUBTRACTED for Cosine.

• Exam Tip 1: Lay out your work very clearly, step-by-step. Don't do multiple calculations in one line. This helps you track your work and helps the examiner award partial marks if you make a small error.

• Exam Tip 2: If you have a scientific calculator, use it to check your final answer. But you must show the full logarithm method to get marks.