Logarithms, Exponentials & Functional Equations

Logarithms and exponentials form the backbone of many JEE problems across algebra, calculus, and coordinate geometry. Mastery of their properties eliminates calculation errors and speeds up problem-solving.

Logarithm Definition: log_a(x) = y means a^y = x, where a > 0, a != 1, x > 0. The logarithm is the inverse of the exponential function.

Fundamental Properties:

• log_a(xy) = log_a(x) + log_a(y) (product rule)
• log_a(x/y) = log_a(x) - log_a(y) (quotient rule)
• log_a(x^n) = n*log_a(x) (power rule)
• log_a(1) = 0, log_a(a) = 1
• a^(log_a(x)) = x (inverse property)

Change of Base Formula: log_a(b) = log_c(b)/log_c(a) = 1/log_b(a). This is the single most useful identity — it converts any logarithm to natural or common log.

Exponential Properties:

• a^m * a^n = a^(m+n), (a^m)^n = a^(mn), (ab)^n = a^n * b^n
• $a^{0}$ = 1 (for a != 0), a^(-n) = 1/a^n
• If a^x = a^y then x = y (for a > 0, a != 1)

Logarithmic Equations: When solving log equations, always check that arguments are positive and bases are valid (positive and not 1). Extraneous solutions are the #1 source of errors.

Exponential Equations: For equations like 2^x + 2^(-x) = 5, substitute t = 2^x to get t + 1/t = 5, yielding a quadratic $t^{2}$ - 5t + 1 = 0.

Functional Equations: Common types tested in JEE:

• f(x+y) = f(x)*f(y) implies f(x) = a^x for some a > 0
• f(x+y) = f(x)+f(y) implies f(x) = kx (with continuity)
• f(xy) = f(x)+f(y) implies f(x) = log_a(x) for some a

Logarithmic Inequalities: The direction of inequality depends on the base:

• If a > 1: log_a(x) > log_a(y) iff x > y
• If 0 < a < 1: log_a(x) > log_a(y) iff x < y (inequality reverses)