Logarithms and Exponents: Laws, Formulas, Log Values and Solved Examples for SSC CGL and CHSL Maths

Exponents (indices) and logarithms are closely related, and questions on them in SSC CGL and CHSL are based on a small set of laws. This guide on logarithms and exponents for SSC covers the laws of indices, the laws of logs, standard log values, finding the number of digits and the usual question types, with solved examples, tricks and practice questions.

Laws of Exponents (Indices)

Definition of Logarithm

If a^x = N, then x = log_a N. For example, 2³ = 8, so log₂ 8 = 3. The base a must be positive and not equal to 1, and N must be positive.

Laws of Logarithms

Standard Values (Base 10)

Number of Digits Using Logs

If log₁₀ N = x, then the number of digits in N = (integer part of x) + 1. For example, 2^20: 20 × 0.3010 = 6.02, so there are 7 digits. 3^10: 10 × 0.4771 = 4.771, so there are 5 digits (3^10 = 59,049).

Solved Examples

Example 1. Find log₂ 64.
64 = 2^6, so the value is 6.

Example 2. Find log 5 + log 20 (base 10).
log(5 × 20) = log 100 = 2.

Example 3. Find log₃ 81 − log₃ 9.
4 − 2 = 2.

Example 4. If log 2 = 0.3010, find log 8 and log 5.
log 8 = 3 × 0.3010 = 0.9030. log 5 = log(10/2) = 1 − 0.3010 = 0.6990.

Example 5. Solve log₂ x = 5.
x = 2^5 = 32.

Example 6. Find x if log_x 625 = 4.
x^4 = 625 = 5^4, so x = 5.

Example 7. Solve 2^(x + 1) = 32.
32 = 2^5, so x + 1 = 5 and x = 4.

Example 8. Solve 9^x = 27.
3^(2x) = 3^3, so 2x = 3 and x = 1.5.

Example 9. Simplify (2^5 × 2^3) / 2^6.
2^(5 + 3 − 6) = 2^2 = 4.

Common Question Types

Tricks

• Learn powers of 2, 3, 5 and 7 up to at least 2^10, 3^6, 5^4 and 7^3.
• Make the bases equal in index equations.
• Remember that log of a number less than 1 is negative.
• Use the change of base formula when bases differ.
• log 1 = 0 and log_a a = 1 in any base.

Quick Tips

• Write the number as a power of the base to find the log.
• Make bases equal in index equations.
• Learn log 2 = 0.3010 and log 3 = 0.4771.
• For the number of digits, add 1 to the integer part of the log.

Quick Revision Points

• a^m × a^n = a^(m + n); (a^m)^n = a^(mn).
• a^0 = 1 and a^(−n) = 1/a^n.
• log(mn) = log m + log n; log(m/n) = log m − log n.
• log(m^n) = n log m.
• Digits in N = integer part of log10 N + 1.

Practice Questions With Answers

1. Find log₃ 243.
   Answer: 5
2. Find log 4 + log 25 (base 10).
   Answer: 2
3. Solve 3^x = 81.
   Answer: 4
4. If log 2 = 0.3010, find log 16.
   Answer: 1.2040
5. Find log₅ 125.
   Answer: 3
6. Solve 4^x = 8.
   Answer: 1.5
7. Find the number of digits in 2^30 (log 2 = 0.3010).
   Answer: 10
8. Find log₂ (1/8).
   Answer: −3
9. Simplify (3^4 × 3^2) / 3^3.
   Answer: 27
10. Find x if log_x 64 = 3.
    Answer: 4

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Quantitative Aptitude notes page, read the previous topic Chain Rule and Variation and the next topic Set Theory and Venn Diagram.

Frequently Asked Questions

What is a logarithm?

If a^x = N, then x is the logarithm of N to the base a, written log_a N.

What is the product law of logarithms?

log(mn) = log m + log n.

What is the quotient law of logarithms?

log(m/n) = log m − log n.

What is the power law of logarithms?

log(m^n) = n times log m.

What is the value of log 1?

It is 0 in any base.

What is log_a a?

It is always 1.

What is the value of log 2 in base 10?

About 0.3010.

How do I find the number of digits of a number using logs?

Find log10 of the number, take the integer part and add 1.

What is the change of base formula?

log_a b = log b / log a.

Can I take the log of a negative number?

No. The log is defined only for positive numbers.

What is a^0?

It is 1 for any non-zero a.

Are logs asked in SSC CGL?

Yes. 1 or 2 questions on logs and indices appear in Tier I and Tier II.