FORMULA LIBERARY : LOGARITHM

LOGARITHM

— JEE MATH APEX —

JEE MATH APEX

1. Basic Definition & Fundamental Identity

• If a^x = N, then x = log_a N  (where a > 0, a ≠ 1, N > 0)
• Fundamental identity: a^{log_a N} = N
• Conditions: Base a > 0, a ≠ 1; Argument N > 0
• Common logarithm (base 10): log₁₀ N = log N
• Natural logarithm (base e): log_e N = ln N

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2. Fundamental Laws of Logarithms

Product Rule:
• log_a(MN) = log_a M + log_a N

Quotient Rule:
• log_a(M/N) = log_a M − log_a N

Power Rule:
• log_a(Mⁿ) = n·log_a M

Root Rule:
• log_a(ⁿ√M) = (1/n)·log_a M

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3. Base Change Rules

• log_a b = (log_c b) / (log_c a)  (Change of base)
• log_a b = 1 / (log_b a)  (Reciprocal property)
• log_a b · log_b a = 1
• log_a b · log_b c · log_c d = log_a d
• log_a b · log_b c · log_c a = 1
• a^{log_b c} = c^{log_b a}

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4. Special Logarithm Values

• log_a 1 = 0
• log_a a = 1
• log_a aⁿ = n
• log_a (1/a) = −1
• log_a (a^x) = x
• log₁₀ 10ⁿ = n  |  log_e eⁿ = n

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5. Exponential & Logarithmic Forms

• Exponential form: a^x = N
• Logarithmic form: x = log_a N
• log_a N = x ⇔ a^x = N
• log_a b = c ⇔ a^c = b

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6. Logarithmic Inequalities

• If a > 1, then log_a M > log_a N ⇔ M > N
• If 0 < a < 1, then log_a M > log_a N ⇔ M < N
• For log_a M > 0: if a>1, M>1; if 0<a<1, 0<M<1
• For log_a M < 0: if a>1, 0<M<1; if 0<a1
• Domain check is essential before solving logarithmic inequalities.

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7. Characteristic & Mantissa

• Every positive number N can be written as: log₁₀ N = Characteristic + Mantissa
• Characteristic: The integer part of log₁₀ N.
• Mantissa: The decimal (fractional) part of log₁₀ N.
• Mantissa is always non-negative (0 ≤ mantissa < 1).
• For N ≥ 1: Characteristic = (Number of digits before decimal − 1)
• For 0 < N < 1: Characteristic is negative; its absolute value = (Number of zeros after decimal before first significant digit + 1)
• Example: log₁₀ 250 = 2 + log₁₀ 2.5 (Characteristic = 2, Mantissa = log 2.5)

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8. Advanced & Special Formulae

• log_a b = log_aⁿ bⁿ
• log_aⁿ b^m = (m/n) · log_a b
• log_a b · log_c d = log_c b · log_a d
• x^{log_a y} = y^{log_a x}
• log_a(b + c) + log_a(b − c) = log_a(b² − c²)
• If x = log_b a, then a = b^x

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9. Series & Summation Using Logs

• log_a(x₁ · x₂ · x₃ · … · x_n) = log_a x₁ + log_a x₂ + … + log_a x_n
• Σ log_a x_i = log_a(Π x_i)
• log_a(xⁿ) = n · log_a x
• Sum of arithmetic series in log form: log_a b + log_a b² + log_a b³ + … = log_a b (1+2+3+…)

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10. Graphs & Key Points

• Domain of log_a x: x ∈ (0, ∞)
• Range of log_a x: (−∞, ∞) i.e., all real numbers
• Graph passes through (1, 0) always
• Graph passes through (a, 1)
• If a > 1: log_a x is strictly increasing
• If 0 < a < 1: log_a x is strictly decreasing
• Y-axis (x = 0) is vertical asymptote