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Logarithm formulas

Master Logarithm through 30 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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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
Q1MCQQuadratic in log
The number of real solutions of $(\log_2 x)^2-\log_2 x-2=0$ is:
- A
  $2$
- B
  $1$
- C
  $0$
- D
  $3$
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
Q1MCQNested logs
The value of $\log_2\big(\log_2(\log_2 16)\big)$ is:
- A
  $1$
- B
  $2$
- C
  $4$
- D
  $0$
Chain relation
$lo g_{a} m = lo g_{b} m \cdot lo g_{a} b$
a, b, m > 0; a, b ≠ 1
Q1MCQChain relation
If $\log_3 5 = a$ and $\log_5 7 = b$, then $\log_3 7$ equals:
- A
  $ab$
- B
  $a+b$
- C
  $\dfrac{a}{b}$
- D
  $\dfrac{b}{a}$
Reciprocal of base
$lo g_{b} a = \frac{1}{lo g _{a} b}$
a, b > 0; a, b ≠ 1
Q1MCQReciprocal + product rule
The value of $\dfrac{1}{\log_2 30}+\dfrac{1}{\log_3 30}+\dfrac{1}{\log_5 30}$ is:
- A
  $1$
- B
  $30$
- C
  $\log_{30}10$
- D
  $0$
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
Q1MCQChange of base
If $\log_{12}27=a$, then $\log_6 16$ equals:
- A
  $\dfrac{4(3-a)}{3+a}$
- B
  $\dfrac{3(4-a)}{4+a}$
- C
  $\dfrac{4(3+a)}{3-a}$
- D
  $\dfrac{3-a}{3+a}$
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
Q1MCQSwap identity
$5^{\log_{10} 2}$ equals:
- A
  $2^{\log_{10}5}$
- B
  $10$
- C
  $2$
- D
  $25$
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).
Q1MCQDomain and range
The domain of $f(x)=\log_2(x-3)$ is:
- A
  $(3,\infty)$
- B
  $(-\infty,3)$
- C
  $[3,\infty)$
- D
  $\mathbb{R}$
Monotonicity
$a > 1 : ↑, 0 < a < 1 : ↓$
Increasing if a > 1; decreasing if 0 < a < 1.
Q1MCQMonotonicity
Which is greater, $\log_2 5$ or $\log_2 7$?
- A
  $\log_2 7$
- B
  $\log_2 5$
- C
  equal
- D
  cannot be decided
Sign of log (positive)
$lo g_{a} x > 0$
if (a > 1, x > 1) or (0 < a < 1, 0 < x < 1)
Q1MCQSign of log (base < 1)
For which values of $x$ is $\log_{0.3}x>0$?
- A
  $(0,1)$
- B
  $(1,\infty)$
- C
  $(0,0.3)$
- D
  $(0.3,1)$
Sign of log (negative)
$lo g_{a} x < 0$
if (0 < a < 1, x > 1) or (a > 1, 0 < x < 1)
Q1MCQSign of log (negative)
$\log_{1/3} 5$ is:
- A
  negative
- B
  positive
- C
  zero
- D
  greater than 1
Bounds when a>1
$0 < lo g_{a} x < 1$
if (a > 1, 1 < x < a) or (0 < a < 1, a < x < 1)
Q1MCQBounds when a>1
If $a>1$ and $1<x<a$, then $\log_a x$ lies in:
- A
  $(0,1)$
- B
  $(1,\infty)$
- C
  $(-1,0)$
- D
  $(-\infty,0)$
Bounds when 0<a<1
$lo g_{a} x > 1$
if (a > 1, x > a) or (0 < a < 1, 0 < x < a)
Q1MCQBounds when 0<a<1
If $0<a<1$ and $0<x<a$, then $\log_a x$ is:
- A
  greater than $1$
- B
  in $(0,1)$
- C
  negative
- D
  equal to $1$
Solving inequality (a>1)
$lo g_{a} x > α ⟺ x > a^{α}$
for a > 1 (and logₐx < α ⇔ 0 < x < a^α)
Q1MCQSolving inequality (a>1)
The solution set of $\log_2 x < 3$ (with $x>0$) is:
- A
  $(0,8)$
- B
  $(8,\infty)$
- C
  $(0,3)$
- D
  $(3,\infty)$
Solving inequality (0<a<1)
$lo g_{a} x > α ⟺ 0 < x < a^{α}$
for 0 < a < 1 (and logₐx < α ⇔ x > a^α)
Q1MCQInequality (base < 1)
The solution set of $\log_{1/2}(x-1)>-1$ is:
- A
  $(1,3)$
- B
  $(3,\infty)$
- C
  $(1,2)$
- D
  $(-\infty,3)$
Types of logarithm
$lo g_{10} x = common, lo g_{e} x = ln x$
base 10 → common (Briggs); base e → natural (Napier)
Q2MCQCharacteristic / magnitude
If $\log_{10}x=2.5$, then $x$ lies in:
- A
  $(100,1000)$
- B
  $(10,100)$
- C
  $(1000,10000)$
- D
  $(1,10)$
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
Q1MCQExpansion of ln(1+x)
The coefficient of $x^{3}$ in the expansion of $\ln(1+x)$ is:
- A
  $\dfrac{1}{3}$
- B
  $-\dfrac{1}{3}$
- C
  $\dfrac{1}{2}$
- D
  $-\dfrac{1}{4}$
Expansion of ln(1-x)
$ln (1 - x) = - (x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \dots)$
−1 ≤ x < 1
Q1MCQExpansion of ln(1-x)
The coefficient of $x^{2}$ in the expansion of $\ln(1-x)$ is:
- A
  $-\dfrac{1}{2}$
- B
  $\dfrac{1}{2}$
- C
  $-1$
- D
  $-\dfrac{1}{3}$
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
Q1MCQExpansion of ln((1+x)/(1-x))
Given $\ln\dfrac{1+x}{1-x}=2\left(x+\dfrac{x^3}{3}+\dfrac{x^5}{5}+\cdots\right)$, the coefficient of $x^{5}$ is:
- A
  $\dfrac{2}{5}$
- B
  $\dfrac{1}{5}$
- C
  $\dfrac{2}{3}$
- D
  $2$
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
Q1MCQAM–GM
For $a,b>1$, the least value of $\log_a b + \log_b a$ is:
- A
  $2$
- B
  $1$
- C
  $0$
- D
  $\sqrt{2}$
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

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

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