Lesson Note
INTRODUCTION
Logarithms are a powerful tool that turns multiplication, division, powers and roots into addition, subtraction and division. In a world where calculations were once done with mechanical calculators or even by hand, logarithm tables provided a quick way to compute these operations without a machine. For students in Nigeria, learning to use a logarithm table not only improves mental arithmetic but also prepares them for real‑world problems such as calculating interest, measuring growth rates, and analysing data in science and economics.
The tables list the logarithms of numbers from 1 to 9.9999 (to four decimal places). By combining the integer part of the number with its fractional part, a student can read off the logarithm of any number between 1 and 10. Using the properties of logarithms, one can then perform complex calculations by simple addition or subtraction of the logged values. This lesson will guide you through the steps of using these tables effectively, with worked examples and practice problems to reinforce the concepts.
MAIN CONTENT
Understanding Logarithms and Their Properties
Logarithms are the inverse operation of exponentiation. For a base $10$ logarithm, $ \log x = y $ means that $10^{y} = x$. The key properties are:
- $ \log(ab) = \log a + \log b $
- $ \log(a/b) = \log a - \log b $
- $ \log(a^{n}) = n \log a $
- $ \log(\sqrt[n]{a}) = \frac{1}{n}\log a $
These properties allow us to use a logarithm table to perform multiplication, division, powers and roots.
Worked Example
Find $ \log 500 $ using the table.
Step 1: Express 500 as $5 \times 10^{2}$
Step 2: $$ \log 500 = \log 5 + \log 10^{2} = \log 5 + 2 $$
Step 3: From the table, $ \log 5 = 0.69897 $
Step 4: $ \log 500 = 0.69897 + 2 = 2.69897 $
Practice Questions
Practice Question 1: Find $ \log 200 $.
Answer: $$ \log 200 = \log 2 + 2 = 0.30103 + 2 = 2.30103 $$.
Practice Question 2: Find $ \log 0.05 $.
Answer: $$ \log 0.05 = \log 5 - 2 = 0.69897 - 2 = -1.30103 $$.
Practice Question 3: Find $ \log 2500 $.
Answer: $$ \log 2500 = \log 25 + 2 = 1.39794 + 2 = 3.39794 $$.
How Logarithm Tables Work
A logarithm table is arranged with the first column showing the leading digit(s) of the number and the second column giving the decimal part of its logarithm. To find the log of a number, locate its leading digit(s) in the first column and read the decimal part from the second column. For numbers greater than 10, add the integer part of the log (i.e., the number of zeros after the decimal point).
Worked Example
Find $ \log 3.1416 $.
Step 1: Identify the leading digits: 3.14.
Step 2: From the table, $ \log 3.14 = 0.49715 $.
Step 3: Since 3.1416 is between 3.14 and 3.15, interpolate:
$$ \log 3.1416 \approx 0.49715 + 0.00001 \times (3.1416-3.14)/0.01 = 0.49715 + 0.00001 = 0.49716 $$.
Step 4: The log is $0.49716$.
Practice Questions
Practice Question 1: Find $ \log 7.2 $.
Answer: $ \log 7.2 = 0.85733 $.
Practice Question 2: Find $ \log 0.003 $.
Answer: $ \log 0.003 = -2.52288 $.
Practice Question 3: Find $ \log 12.5 $.
Answer: $ \log 12.5 = 1.09691 $.
Using Log Tables for Multiplication
To multiply two numbers $a$ and $b$, use $ \log(ab) = \log a + \log b $. Add their logarithms, then find the antilogarithm (10 raised to the sum) to obtain the product.
Worked Example
Multiply $23.5$ by $4.2$.
Step 1: $ \log 23.5 = 1.37121 $ (from table).
Step 2: $ \log 4.2 = 0.62325 $.
Step 3: Sum: $1.37121 + 0.62325 = 1.99446$.
Step 4: Antilog: $10^{1.99446} \approx 98.5$ (using table or calculator).
Practice Questions
Practice Question 1: Multiply $15.3$ by $2.7$.
Answer: $41.3$.
Practice Question 2: Multiply $8.4$ by $9.6$.
Answer: $80.6$.
Practice Question 3: Multiply $12.5$ by $0.8$.
Answer: $10.0$.
Using Log Tables for Division
To divide $a$ by $b$, use $ \log(a/b) = \log a - \log b $. Subtract the logarithms and then exponentiate to get the quotient.
Worked Example
Divide $45.6$ by $3.4$.
Step 1: $ \log 45.6 = 1.65930 $.
Step 2: $ \log 3.4 = 0.53148 $.
Step 3: Difference: $1.65930 - 0.53148 = 1.12782$.
Step 4: Antilog: $10^{1.12782} \approx 13.4$.
Practice Questions
Practice Question 1: Divide $18.9$ by $2.3$.
Answer: $8.2$.
Practice Question 2: Divide $50.0$ by $5.0$.
Answer: $10.0$.
Practice Question 3: Divide $7.2$ by $0.9$.
Answer: $8.0$.
Using Log Tables for Finding Powers
To find $a^{n}$, use $ \log(a^{n}) = n \log a $. Multiply the logarithm of $a$ by $n$, then exponentiate.
Worked Example
Find $7^{3}$.
Step 1: $ \log 7 = 0.84510 $.
Step 2: Multiply by 3: $3 \times 0.84510 = 2.53530$.
Step 3: Antilog: $10^{2.53530} \approx 343$.
Practice Questions
Practice Question 1: Find $4^{4}$.
Answer: $256$.
Practice Question 2: Find $9^{2.5}$.
Answer: $9^{2.5} \approx 243$.
Practice Question 3: Find $2^{10}$.
Answer: $1024$.
Using Log Tables for Finding Roots
To find the $n^{th}$ root of $a$, use $ \log(\sqrt[n]{a}) = \frac{1}{n}\log a $. Divide the logarithm by $n$, then exponentiate.
Worked Example
Find the cube root of $64$.
Step 1: $ \log 64 = 1.80618 $.
Step 2: Divide by 3: $1.80618 / 3 = 0.60206$.
Step 3: Antilog: $10^{0.60206} \approx 4$.
Practice Questions
Practice Question 1: Find the square root of $81$.
Answer: $9$.
Practice Question 2: Find the fourth root of $16$.
Answer: $2$.
Practice Question 3: Find the fifth root of $32$.
Answer: $2$.
Applications of Logarithms in Real Life
Logarithms simplify calculations in finance, science and engineering. In capital markets, the compound interest formula $A = P(1 + r/n)^{nt}$ can be evaluated using logs: $ \log A = \log P + nt \log(1 + r/n) $. In population studies, exponential growth models use natural logs to determine growth rates. In physics, logarithmic scales describe sound intensity and earthquake magnitude.
Worked Example
Compute $1.05^{10}$ (10‑year growth at 5% interest).
Step 1: $ \log 1.05 = 0.02119 $.
Step 2: Multiply by 10: $0.21190$.
Step 3: Antilog: $10^{0.21190} \approx 1.629$.
Step 4: The amount grows by a factor of $1.629$.
Practice Questions
Practice Question 1: Compute $1.03^{12}$.
Answer: $1.425$.
Practice Question 2: Compute $1.07^{5}$.
Answer: $1.402$.
Practice Question 3: Compute $1.02^{20}$.
Answer: $1.485$.
Common Errors and Tips for Accuracy
When using logarithm tables, common mistakes include misreading the decimal part, forgetting to add the integer part for numbers >10, and rounding too early. Always double‑check the table entry, keep the full precision of the log during intermediate steps, and only round the final result. Using a calculator to verify a few entries can help build confidence.
Worked Example
Suppose a student reads $ \log 2.5 $ as $0.39794$ (correct) but mistakenly uses $0.39794$ for $ \log 2.6 $. The error propagates:
Step 1: Correct $ \log 2.6 = 0.41497 $.
Step 2: Using $0.39794$ gives $10^{0.39794} \approx 2.5$ instead of $2.6$.
Step 3: The product $5.5 \times 3.3$ would be miscalculated as $18.15$ instead of $18.15$ (close but shows the effect of small errors).
Practice Questions
Practice Question 1: Identify the error in calculating $5.5 \times 3.3$ using $ \log 5.5 = 0.74036 $ and $ \log 3.3 = 0.51851 $ but mistakenly adding 0.74036 + 0.51851 = 1.25887 then taking antilog as $18.15$ (correct).
Answer: No error; the calculation is correct.
Practice Question 2: Correct the result if $ \log 5.5 $ was mistakenly taken as $0.70000$.
Answer: Sum $0.70000 + 0.51851 = 1.21851$, antilog $10^{1.21851} \approx 16.5$.
Practice Question 3: Explain why rounding $ \log 4.2 $ to $0.62$ instead of $0.62325$ changes the product $23.5 \times 4.2$ by about $0.5$.
SUMMARY
Logarithm tables transform complex multiplication, division, powers and roots into simple addition and subtraction. By mastering the table, students can perform accurate calculations without a calculator, a skill that remains valuable in many scientific and financial contexts. Understanding the underlying properties of logarithms also builds a foundation for higher mathematics and real‑world problem solving. Practising with real numbers and checking for common pitfalls will ensure confidence and precision in using these tables.
EVALUATION
1. Compute $ \log 6.3 $.
2. Multiply $12.4$ by $3.8$ using logarithm tables.
3. Divide $54.0$ by $6.0$ using logarithm tables.
4. Find $9^{3}$ using logarithm tables.
5. Find the square root of $49$ using logarithm tables.