Logarithm Rules for Placement Aptitude Tests

Logarithm questions appear in TCS NQT, Infosys, and Wipro placement aptitude sections, and every variant you will encounter traces back to six base rules.

What a Logarithm Is

The formal definition: if a^x = N, then log_a(N) = x. Read as “log of N to the base a equals x.” The base a is a positive number other than 1, and N is positive. Translating that to plain English: to what power must a be raised to get N? That question is how to approach every log problem in a placement test.

A concrete example: 4^3 = 64, so log_4(64) = 3. The base is 4, the result is 64, the exponent is 3. Placement questions flip any one of those three quantities and ask you to find it.

Wikipedia’s logarithm article covers the full formal derivation. For placement purposes, the definition above and the six rules below are all that is needed.

Two notational points that save time in placement tests: when no base is written, assume base 10. When you see ln, the base is e (approximately 2.718), and that notation appears in calculus or data science, not in standard placement aptitude.

Common Logarithm and Natural Logarithm

• Common logarithm uses base 10. Default notation: log(N) without a subscript. log(100) = 2 because 10^2 = 100.
• Natural logarithm uses base e (approximately 2.718). Notation: ln(N). Rarely tested in placement aptitude; appears in probability and advanced maths contexts.

Characteristic and Mantissa

For any log_10(N), the result splits into two parts:

• Characteristic = the integer part (the part to the left of the decimal point)
• Mantissa = the decimal part (always non-negative, always less than 1)

The digit-count rule follows directly:

Number of digits in a positive integer N = characteristic of log_10(N) + 1

Example: log_10(5000) = log_10(5) + log_10(1000) = 0.699 + 3 = 3.699. Characteristic = 3. Digits = 3 + 1 = 4. Correct: 5000 has four digits.

Khan Academy’s introduction to logarithms provides interactive exercises on this and on the rules below.

The Six Core Logarithm Rules

All six rules derive from the definition a^x = N implies log_a(N) = x. Each one can be re-derived in under a minute if you forget it during an exam.

Rule	Formula	Quick check
Identity	log_a(a) = 1	log_5(5) = 1 because 5^1 = 5
Zero	log_a(1) = 0	log_7(1) = 0 because 7^0 = 1
Product	log_a(m*n) = log_a(m) + log_a(n)	log(50) + log(2) = log(100) = 2
Quotient	log_a(m/n) = log_a(m) - log_a(n)	log_5(625) - log_5(25) = log_5(25) = 2
Power	log_a(m^p) = p * log_a(m)	log_2(32) = log_2(2^5) = 5
Change of base	log_a(m) = log_b(m) / log_b(a)	log_4(8) = log_2(8) / log_2(4) = 3/2

A useful corollary of change of base: log_a(b) = 1 / log_b(a). This appears in questions that chain two logarithms and ask for their product.

Counting Digits with Logarithms

The digit-counting question is the most predictable log question in placement aptitude. The method is always the same:

• Step 1: Express the number as a power, e.g., 2^10.
• Step 2: Apply the power rule: log_10(2^10) = 10 * log_10(2) = 10 * 0.301 = 3.01.
• Step 3: Digits = floor(3.01) + 1 = 3 + 1 = 4.
• Step 4 (verify): 2^10 = 1024, which has 4 digits. Correct.

Three anchor values to memorise: log_10(2) = 0.301, log_10(3) = 0.477, log_10(7) = 0.845. Every digit-counting placement question is built around these or around clean powers of 10, 5, or 2.

Note: log_10(5) derives from the anchor values without any additional memorisation: log_10(5) = log_10(10/2) = 1 - 0.301 = 0.699. This covers digit-counting for powers of 5 without adding a fourth value to memorise.

Two more worked examples:

• Digits in 2^20:

  • log_10(2^20) = 20 * 0.301 = 6.02
  • Digits = floor(6.02) + 1 = 7
  • Verify: 2^20 = 1,048,576 (7 digits). Correct.

• Digits in 3^10:

  • log_10(3^10) = 10 * 0.477 = 4.77
  • Digits = floor(4.77) + 1 = 5
  • Verify: 3^10 = 59049 (5 digits). Correct.

• Digits in 5^15:

  • log_10(5^15) = 15 * log_10(5) = 15 * 0.699 = 10.485
  • Digits = floor(10.485) + 1 = 11
  • Verify: 5^15 = 30,517,578,125 (11 digits). Correct.

Logarithm Questions in Placement Aptitude

TCS NQT, Infosys Aptitude Online, and Wipro NLTH all include logarithm problems in their quantitative reasoning sections. Three question types cover the full range.

Type 1: Evaluate the Expression

Find the value directly by converting to exponential form.

• Example: Find log_8(64).
• Method: To what power must 8 be raised to get 64? 8^2 = 64, so log_8(64) = 2. Verify: 8 * 8 = 64. Correct.

Type 2: Simplify Using Rules

Apply one or more of the six rules to collapse the expression to a single number.

• Example: Find the value of log(50) + log(2).

• Method: Product rule. log(50 * 2) = log(100) = 2. Verify: 10^2 = 100. Correct.

• Example: Find log_5(625) - log_5(25).

• Method: Quotient rule. log_5(625 / 25) = log_5(25) = log_5(5^2) = 2. Verify: 625 / 25 = 25 and 5^2 = 25. Correct.

Type 3: Digit Counting

Already covered in the previous section. The one variation to watch for: some questions give log_10(2) = 0.301 as a stated given, and some expect you to recall it. A less common variant asks about a product of two powers: how many digits does 2^10 * 5^10 have? The trick is recognising that 2^10 * 5^10 = (2 * 5)^10 = 10^10, so log_10(10^10) = 10 and digits = 11. Combined-exponent forms of this type appear in both Infosys and TCS NQT.

Logarithms sit in the same quantitative cluster as calendar problems and clock problems: a small rule set, memorised cold, solves the entire category. The arithmetic that follows the log step benefits from the same quick number tricks that help in other quantitative sections.

From Digit Counts to Language Model Perplexity

The power rule that counts digits in 2^10 is the same algebra that defines perplexity in language model evaluation. When a model assigns a per-token log-probability of log_2(1/1024) = -10 bits, its perplexity is 2^10 = 1024. That number measures how surprised the model is by real text. Lower perplexity, better predictions. The arithmetic is identical to the digit-counting method in the previous section.

Shannon entropy (the mathematical measure of information), cross-entropy loss (the training objective for every language model), and the log-probability scores an API returns all run on the product, power, and change-of-base rules in the table above.

TinkerLLM gives you direct access to LLM API calls for ₹299. The model responses include log-probability scores for each token. Reading those scores and understanding what a log-probability of -3.2 means in terms of model confidence is the same numerical literacy this article builds. The six-rule foundation you practised for placement aptitude is the same one that shows up in every model evaluation report.