Logarithms for JEE Main: Properties & Applications

Logarithms are tested directly in JEE Main through one to two questions per session and appear implicitly in sequences and series, calculus (ln x derivative and integration), complex numbers (polar form), and even the Nernst equation in chemistry. A thorough understanding of logarithm properties eliminates an entire category of algebraic errors in exams.

Fundamental Properties

Definition: log_b(a) = x means b^x = a. Conventions: log without base means log₁₀ (common logarithm); ln means log_e (natural logarithm). Essential properties: log(mn) = log m + log n; log(m/n) = log m − log n; log(m^n) = n·log m; log_b(b) = 1; log_b(1) = 0; log_b(0) is undefined; log_b(b^x) = x; b^(log_b x) = x. Change of base: log_a(b) = log_c(b)/log_c(a) = ln b/ln a. Reciprocal: log_a(b) = 1/log_b(a).

JEE Main tests these properties in simplification chains: compute log₅(8) × log₆(5) × log₃(6). Using change of base: = (ln 8/ln 5) × (ln 5/ln 6) × (ln 6/ln 3) = ln 8/ln 3 = log₃(8) = log₃(2³) = 3log₃(2). This telescoping pattern appears frequently. For the sequence series connection, see our Sequences and Series Guide.

Logarithmic Equations

Step 1: identify the domain (argument of log must be > 0; base must be > 0, base ≠ 1). Step 2: apply log properties to simplify. Step 3: convert to algebraic equation and solve. Step 4: verify solution is in the domain. Example: log₂(x² − 3x) = 2. Domain: x² − 3x > 0 → x < 0 or x > 3. Equation: x² − 3x = 2² = 4 → x² − 3x − 4 = 0 → (x−4)(x+1) = 0 → x = 4 or x = −1. Check domain: x=4 gives 16−12=4>0 ✓; x=−1 gives 1+3=4>0 ✓. Both valid. JEE Main frequently sets up equations where one solution violates the domain — always check. Take a free mock test on algebra including logarithms to practise domain checking.

Logarithmic Inequalities

log_b(f(x)) > log_b(g(x)) does NOT always mean f(x) > g(x). The direction depends on the base: if b > 1, the log function is increasing, so log_b(f) > log_b(g) ↔ f > g. If 0 < b < 1, the log function is decreasing, so log_b(f) > log_b(g) ↔ f < g. This sign reversal for bases between 0 and 1 is the most common source of errors in logarithmic inequality problems. JEE Main deliberately uses base 1/2, 1/3 to catch students who forget to flip the inequality.

Systematic approach to log_a(f(x)) > c: Case 1 (a > 1): f(x) > a^c and f(x) > 0 (domain). Case 2 (0 < a < 1): f(x) < a^c and f(x) > 0 (domain). Combine with the domain restriction for the complete solution.

Graph of Logarithmic Function

y = log_b(x) is defined only for x > 0. For b > 1: increasing function; log_b(x) > 0 for x > 1, log_b(x) < 0 for 0 < x < 1; passes through (1, 0) and (b, 1). For 0 < b < 1: decreasing function; log_b(x) < 0 for x > 1. The graphs of y = log_b(x) and y = b^x are reflections in the line y = x. JEE Main tests graph-based questions: which of the following is the graph of log_(1/2)(x)?

Natural Logarithm and Calculus Connection

d/dx[ln x] = 1/x. ∫(1/x)dx = ln|x| + C. ∫ln x dx = x·ln x − x + C. ln e = 1; ln 1 = 0. Limits: lim(x→0) ln(1+x)/x = 1; lim(x→∞) ln x / x = 0. JEE Main uses ln in the context of integration (∫dx/(ax+b)) and in differential equations (variable separable type that involves ln). For the full integration chapter, see our Integration Techniques Guide.

Exam Strategy

Logarithm questions in JEE Main fall into three types: (1) property application/simplification — chain telescoping; (2) equation solving — domain check is critical; (3) inequality — base > 1 vs. 0 < base < 1 determines inequality direction. Master these three types and logarithm questions become fast guaranteed marks. The domain check step is dropped by 30-40% of students — never skip it. Upgrade for ₹149/month for 100+ logarithm problems with domain analysis and full solutions.