Applications of Logarithms

Logarithms make difficult calculations easier — especially when multiplying, dividing, or working with powers and roots. Before calculators were widely used, log tables helped students and scientists perform these operations quickly and accurately.

 

Using Logarithms for Multiplication

To multiply two numbers using logs:

1. Find the logarithm of each number.
2. Add the logs together.
3. Find the antilog of the result.

Example: Multiply 25.6 by 3.42 using logs.

Step 1: Use log tables to find the logs:

• $\log_{10}(25.6) \approx 1.4082$
• $\log_{10}(3.42) \approx 0.5340$

Step 2: Add the logs:

$$1.4082 + 0.5340 = 1.9422$$

Step 3: Find the antilog of 1.9422:

$$\text{Antilog}(1.9422) \approx 87.6$$

So, $25.6 \times 3.42 \approx 87.6$

 

Using Logarithms for Division

To divide two numbers using logs:

1. Find the logarithm of the numerator.
2. Subtract the logarithm of the denominator.
3. Find the antilog of the result.

Example: Divide 245 by 5.2 using logs.

• $\log_{10}(245) \approx 2.3892$
• $\log_{10}(5.2) \approx 0.7160$

Subtract:

$$2.3892 - 0.7160 = 1.6732$$

Antilog of 1.6732 is approximately 47.2

So, $245 \div 5.2 \approx 47.2$

 

Using Logs for Powers

To find $a^b$ using logs:

1. Find $\log_{10}(a)$
2. Multiply the log by $b$
3. Find the antilog of the result

Example: Find $6.4^3$

• $\log_{10}(6.4) \approx 0.8062$
• Multiply: $0.8062 \times 3 = 2.4186$
• Antilog of 2.4186 is approximately 262

So, $6.4^3 \approx 262$

 

Using Logs for Roots

To find roots using logs:

1. Take the log of the number.
2. Divide the log by the root (e.g., 2 for square root, 3 for cube root).
3. Find the antilog of the result.

Example: Find $\sqrt[3]{326}$

• $\log_{10}(326) \approx 2.5132$
• $2.5132 \div 3 = 0.8377$
• Antilog of 0.8377 is approximately 6.88

So, $\sqrt[3]{326} \approx 6.88$

 

Working with Log Tables

WAEC sometimes gives a log/antilog table (Table F). Here's how to use them:

• Log Table: Use the first 2 digits + the 3rd as a column. E.g., for 3.42 → use row 34, column 2.
• Antilog Table: Reverse the process — locate the decimal part of your log and find the corresponding number.

Split logs like this: $\log_{10}(245) = 2.3892$, where 2 is the characteristic, and 0.3892 is the mantissa.