Logarithm formulas

Every formula JEE Advanced expects you to know for Logarithm, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 30
Subject: Mathematics
Exam: JEE Advanced
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Logarithm, every formula

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Definition of logarithm
$lo g_{a} b = x ⟺ a^{x} = b$
a, b > 0 and a ≠ 1
Log of its own base
$lo g_{a} a = 1$
a > 0, a ≠ 1
Log of one
$lo g_{a} 1 = 0$
a > 0, a ≠ 1
Anti-log identity
$a^{l o g_{a} m} = m$
a, m > 0, a ≠ 1
Product rule
$lo g_{a} (mn) = lo g_{a} m + lo g_{a} n$
a, m, n > 0, a ≠ 1
Quotient rule
$lo g_{a} \frac{m}{n} = lo g_{a} m - lo g_{a} n$
a, m, n > 0, a ≠ 1
Power rule
$lo g_{a} m^{n} = n lo g_{a} m$
a, m > 0, a ≠ 1; n real
Chain relation
$lo g_{a} m = lo g_{b} m \cdot lo g_{a} b$
a, b, m > 0; a, b ≠ 1
Reciprocal of base
$lo g_{b} a = \frac{1}{lo g _{a} b}$
a, b > 0; a, b ≠ 1
Change of base
$lo g_{a} m = \frac{lo g _{b} m}{lo g _{b} a} = \frac{lo g m}{lo g a}$
Change of base. a, b, m > 0; a, b ≠ 1
Base-power rule
$lo g_{a^{b}} m^{n} = \frac{n}{b} lo g_{a} m$
a, m > 0, a ≠ 1, b ≠ 0
Swap identity
$a^{l o g_{x} b} = b^{l o g_{x} a}$
a, b, x > 0, x ≠ 1
Domain and range
$Domain = (0, \infty), Range = R$
y = logₐx: monotonic & continuous on (0, ∞); mirror image of aˣ about y = x; vertical asymptote x = 0; graph passes through (1, 0).
Monotonicity
$a > 1 : ↑, 0 < a < 1 : ↓$
Increasing if a > 1; decreasing if 0 < a < 1.
Sign of log (positive)
$lo g_{a} x > 0$
if (a > 1, x > 1) or (0 < a < 1, 0 < x < 1)
Sign of log (negative)
$lo g_{a} x < 0$
if (0 < a < 1, x > 1) or (a > 1, 0 < x < 1)
Bounds when a>1
$0 < lo g_{a} x < 1$
if (a > 1, 1 < x < a) or (0 < a < 1, a < x < 1)
Bounds when 0<a<1
$lo g_{a} x > 1$
if (a > 1, x > a) or (0 < a < 1, 0 < x < a)
Solving inequality (a>1)
$lo g_{a} x > α ⟺ x > a^{α}$
for a > 1 (and logₐx < α ⇔ 0 < x < a^α)
Solving inequality (0<a<1)
$lo g_{a} x > α ⟺ 0 < x < a^{α}$
for 0 < a < 1 (and logₐx < α ⇔ x > a^α)
Types of logarithm
$lo g_{10} x = common, lo g_{e} x = ln x$
base 10 → common (Briggs); base e → natural (Napier)
Expansion of ln(1+x)
$ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \frac{x ^{4}}{4} + \dots$
−1 < x ≤ 1
Expansion of ln(1-x)
$ln (1 - x) = - (x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \dots)$
−1 ≤ x < 1
Expansion of ln((1+x)/(1-x))
$ln \frac{1 + x}{1 - x} = 2 (x + \frac{x ^{3}}{3} + \frac{x ^{5}}{5} + \dots)$
|x| < 1
Cyclic product of logs
$lo g_{a} b \cdot lo g_{b} c \cdot lo g_{c} a = 1$
a, b, c > 0 and ≠ 1
Telescoping chain
$lo g_{a} b \cdot lo g_{b} c \cdot lo g_{c} d = lo g_{a} d$
each base > 0 and ≠ 1
Sum with its reciprocal (AM–GM)
$lo g_{a} b + lo g_{b} a \geq 2$
for a, b > 1 (or both in (0,1)); equality when a = b
Number of digits (characteristic)
$Digits of N = ⌊ lo g_{10} N ⌋ + 1$
for N ≥ 1 (common logarithm)
Leading zeros for 0 < N < 1
$Zeros = - ⌊ lo g_{10} N ⌋ - 1$
zeros after the decimal point before the first significant figure
Log of a power of the base
$lo g_{a} a^{x} = x$
a > 0, a ≠ 1

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Logarithm formulas

Logarithm, every formula

Definition of logarithm

Log of its own base

Log of one

Anti-log identity

Product rule

Quotient rule

Power rule

Chain relation

Reciprocal of base

Change of base

Base-power rule

Swap identity

Domain and range

Monotonicity

Sign of log (positive)

Sign of log (negative)

Bounds when a>1

Bounds when 0<a<1

Solving inequality (a>1)

Solving inequality (0<a<1)

Types of logarithm

Expansion of ln(1+x)

Expansion of ln(1-x)

Expansion of ln((1+x)/(1-x))

Cyclic product of logs

Telescoping chain

Sum with its reciprocal (AM–GM)

Number of digits (characteristic)

Leading zeros for 0 < N < 1

Log of a power of the base

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