Logarithm - JEE Advanced Previous Year Questions with Solutions

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Why Logarithm Matters in JEE Advanced 

Logarithm is one of the foundational chapters in JEE Advanced Mathematics. It is not an isolated topic — logarithmic manipulation appears directly inside problems from inequalities, functions, sequences, and even complex numbers.

JEE Advanced regularly tests at a level well above what students typically prepare in Class 11. A question may combine log inequalities with domain restrictions, or require simplification across multiple bases simultaneously. Students who treat logarithms as a formula-recall topic — rather than a reasoning topic — consistently lose marks here.

How Often Does a Logarithm Appear in JEE Advanced?

Source: JEE Advanced official question papers. The topic appears consistently across years, making it a reliable scoring opportunity.

💡 Expert Tip by Saransh Gupta, IIT Bombay AIR-41: "Logarithm in JEE Advanced is never purely computational. Every log question has a thinking step — either a domain check, a base comparison, or a substitution trick. Students who skip that step lose the question even when their algebra is correct."

JEE Advanced Previous Year Questions — Logarithm 

Key Logarithm Properties You Must Know 

Before attempting any JEE Advanced PYQ on logarithms, these properties must be second nature — not just memorised, but understood well enough to apply in reverse or in combination.

The 7 Core Logarithm Identities for JEE Advanced

Domain Conditions — The #1 Source of Errors

For log_a(x) to be defined:

• x > 0 (argument must be strictly positive)
• a > 0 and a ≠ 1 (base must be positive and not equal to 1)

In JEE Advanced, forgetting to apply domain conditions after solving a log equation is the most common reason for a correct algebraic answer being marked wrong. Always verify your solution satisfies the domain before writing the final answer.

Logarithm Inequalities — Critical Rule

When solving log_a(f(x)) > log_a(g(x)):

• If a > 1: the inequality direction is preserved → f(x) > g(x)
• If 0 < a < 1: the inequality direction is reversed → f(x) < g(x)

This rule reversal is the single most tested concept in JEE Advanced log inequality problems.

Common Mistakes Students Make in Log Questions 

Understanding where marks are lost is as valuable as knowing how to solve correctly.

Mistake 1 — Skipping Domain Verification

After solving a log equation and finding x = −3 or x = 0, students often write this as the final answer without checking if it satisfies the domain condition (argument > 0, base ≠ 1). In JEE Advanced, this costs the full mark for the question.

Fix: Always write "Domain check:" as a mandatory final step before boxing your answer.

Mistake 2 — Forgetting Base Comparison in Inequalities

When solving log_a(f) > log_a(g), students drop the inequality direction without checking whether a > 1 or 0 < a < 1. This produces the exact wrong answer — opposite to the correct one.

Fix: Write the base value explicitly before solving any log inequality. Confirm: "a = 0.5, so a < 1, direction reverses."

Mistake 3 — Treating log_a(b) and log_b(a) as Equal

These are reciprocals, not equal values. log₂(8) = 3 but log₈(2) = 1/3.

Fix: Practise change of base until the reciprocal relationship is reflexive, not recalled.

Mistake 4 — Errors in Change of Base During Nested Expressions

In problems like Question 2 above, students convert bases incorrectly when the expression is nested. Using a consistent target base (always convert to ln or log₂) prevents arithmetic errors.

How to Approach Logarithm Questions in JEE Advanced 

Step-by-Step Method for Any Log Problem

1. Identify the question type — Is it an equation, inequality, simplification, or nested expression?
2. State domain conditions upfront — Write constraints before solving, not after.
3. Choose a consistent base — Convert all logs to a single base (usually ln or log base 2) to avoid errors.
4. Apply the relevant identity — From the 7 properties listed above, pick the one that simplifies the expression.
5. Solve algebraically — Use substitution (let log_a(x) = t) for cleaner working.
6. Verify against domain — Cross-check every solution against the domain constraints from Step 2.

How Much Time to Spend on a Log Question in JEE Advanced?

If a log question is taking longer than 7 minutes in the exam, mark it and move on. Return to it after completing other questions — do not let one question consume time from the rest of the paper.

What Level of Difficulty Does JEE Advanced Set for Logarithm?

Logarithm in JEE Advanced is typically at the moderate-to-hard level. Questions rarely test only one property. A typical Advanced log question combines:

• Change of base with an inequality
• A domain restriction that eliminates one solution
• Or a substitution that converts a log equation into a quadratic

Practising only JEE Main-level log questions is insufficient preparation. You need JEE Advanced PYQs and module-level problems where the answer is not immediately obvious.

For structured practice with Advanced-difficulty logarithm exercises and faculty-guided solutions, eSaral's JEE course modules include topic-wise question banks specifically built to this standard.