Class 12 Maths Formula Sheet

The Class 12 Mathematics curriculum is intensive and demanding, serving as a gateway to higher education in engineering, sciences, and commerce. Success at this level requires absolute fluency in a vast number of complex formulas. To aid your preparation, we have compiled this exhaustive Class 12 Maths formula sheet. It encompasses all the critical equations across advanced topics like algebra, calculus, vectors, probability, trigonometry, and logarithms. From complex integrals to 3D geometry formulas, everything you need is organised here for quick reference. Mastering these formulas is non-negotiable if you want to excel in your board exams and competitive entrance tests. Make this sheet an integral part of your daily study routine, use it to verify your solutions, and ensure you are completely exam-ready.

algebra

Quadratic Formula

Class 10algebra

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra

a^2 - b^2 = (a - b)(a + b)

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra

a_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra

S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra

a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra

S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra

S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

Rewriting quadratic into vertex form

#quadratic#factoring

calculus

Power Rule (Derivative)

Calculuscalculus

\frac{d}{dx}[x^n] = n x^{n-1}

Derivative of a power function

#derivative#power

Product Rule

Calculuscalculus

\frac{d}{dx}[u \cdot v] = u'v + uv'

Derivative of product of two functions

#derivative#product

Quotient Rule

Calculuscalculus

\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}

Derivative of quotient of two functions

#derivative#quotient

Chain Rule

Calculuscalculus

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of composite functions

#derivative#chain

Derivative of Exponential

Calculuscalculus

\frac{d}{dx}[e^x] = e^x, \quad \frac{d}{dx}[a^x] = a^x \ln a

Derivative of natural and base-a exponential functions

#derivative#exponential

Derivative of Logarithm

Calculuscalculus

\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of natural and base-a logarithms

#derivative#logarithm

Standard Trig Derivatives

Calculuscalculus

\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin x, \quad \frac{d}{dx}[\tan x] = \sec^2 x

Derivatives of basic trigonometric functions

#derivative#trigonometry

Power Rule (Integral)

Calculuscalculus

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Indefinite integral of power function

#integral#power

Reciprocal Integral

Calculuscalculus

\int \frac{1}{x} \, dx = \ln|x| + C

Integral of 1/x yielding natural logarithm

#integral#logarithm

Integration by Parts

Calculuscalculus

\int u \, dv = uv - \int v \, du

Standard integration technique for products

#integral#parts

Fundamental Theorem of Calculus

Calculuscalculus

\int_a^b f(x) \, dx = F(b) - F(a)

Evaluating definite integral from antiderivative F(x)

#integral#theorem

L'Hôpital's Rule

Calculuscalculus

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

Evaluating indeterminate forms (0/0 or ∞/∞)

#limit#lhopital

vectors

Vector Dot Product

vectors

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z

Scalar product of two vectors

#vectors#dot-product

Vector Cross Product

vectors

\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

Vector product yielding orthogonal vector

#vectors#cross-product

Vector Magnitude

vectors

|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Euclidean norm or length of a vector

#vectors#magnitude

Unit Vector

vectors

\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of length 1 pointing in same direction

#vectors#unit-vector

Scalar Projection of A onto B

vectors

\text{proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Magnitude of vector a in direction of vector b

#vectors#projection

Matrix Determinant (2x2)

vectors

\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

Determinant of a 2 by 2 matrix

#matrices#determinant

Matrix Inverse (2x2)

vectors

\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Inverse of an invertible 2 by 2 matrix

#matrices#inverse

Characteristic Equation (Eigenvalues)

vectors

\det(A - \lambda I) = 0

Equation for finding eigenvalues λ of matrix A

#matrices#eigenvalues

probability

Permutations Formula

probability

P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

probability

C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

probability

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

probability

P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

probability

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

probability

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

probability

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

probability

E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

probability

\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

probability

f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

trigonometry

Pythagorean Identity

Class 10trigonometry

\sin^2\theta + \cos^2\theta = 1

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry

1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry

1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry

\sin(2\theta) = 2\sin\theta\cos\theta

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry

\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#sines

Law of Cosines

Class 10trigonometry

c^2 = a^2 + b^2 - 2ab\cos C

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#cosines

logarithms

Product Rule for Logs

logarithms

\log_b(xy) = \log_b(x) + \log_b(y)

Logarithm of a product equals sum of logarithms

#product#logarithm

Quotient Rule for Logs

logarithms

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

Logarithm of a quotient equals difference of logarithms

#quotient#logarithm

Power Rule for Logs

logarithms

\log_b(x^k) = k \log_b(x)

Exponent comes out as a multiplier

#power#exponent#logarithm

Change of Base Formula

logarithms

\log_b(x) = \frac{\log_k(x)}{\log_k(b)} = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Converting logarithm to a different base k

#base#conversion

Identity & Zero Properties

logarithms

\log_b(b) = 1, \quad \log_b(1) = 0

Basic identity properties of logarithms

#identity#logarithm

Inverse Properties

logarithms

b^{\log_b x} = x, \quad \log_b(b^x) = x

Logarithms and exponentials are inverse functions

#inverse#exponential

Natural Log & Euler Constant

logarithms

\ln(x) = \log_e(x), \quad e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Definition of natural logarithm and base e

#natural#euler

Exponential Laws

logarithms

a^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

Fundamental laws of exponent arithmetic

#exponential#powers

Related Formula Sheets

Class 10 Maths Formula Sheet A-Level Maths Formula Sheet

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Class 12 Maths Formula Sheet

algebra

Quadratic Formula

Class 10algebra

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra

a^2 - b^2 = (a - b)(a + b)

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra

a_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra

S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra

a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra

S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra

S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

Rewriting quadratic into vertex form

#quadratic#factoring

calculus

Power Rule (Derivative)

Calculuscalculus

\frac{d}{dx}[x^n] = n x^{n-1}

Derivative of a power function

#derivative#power

Product Rule

Calculuscalculus

\frac{d}{dx}[u \cdot v] = u'v + uv'

Derivative of product of two functions

#derivative#product

Quotient Rule

Calculuscalculus

\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}

Derivative of quotient of two functions

#derivative#quotient

Chain Rule

Calculuscalculus

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of composite functions

#derivative#chain

Derivative of Exponential

Calculuscalculus

\frac{d}{dx}[e^x] = e^x, \quad \frac{d}{dx}[a^x] = a^x \ln a

Derivative of natural and base-a exponential functions

#derivative#exponential

Derivative of Logarithm

Calculuscalculus

\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of natural and base-a logarithms

#derivative#logarithm

Standard Trig Derivatives

Calculuscalculus

\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin x, \quad \frac{d}{dx}[\tan x] = \sec^2 x

Derivatives of basic trigonometric functions

#derivative#trigonometry

Power Rule (Integral)

Calculuscalculus

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Indefinite integral of power function

#integral#power

Reciprocal Integral

Calculuscalculus

\int \frac{1}{x} \, dx = \ln|x| + C

Integral of 1/x yielding natural logarithm

#integral#logarithm

Integration by Parts

Calculuscalculus

\int u \, dv = uv - \int v \, du

Standard integration technique for products

#integral#parts

Fundamental Theorem of Calculus

Calculuscalculus

\int_a^b f(x) \, dx = F(b) - F(a)

Evaluating definite integral from antiderivative F(x)

#integral#theorem

L'Hôpital's Rule

Calculuscalculus

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

Evaluating indeterminate forms (0/0 or ∞/∞)

#limit#lhopital

vectors

Vector Dot Product

vectors

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z

Scalar product of two vectors

#vectors#dot-product

Vector Cross Product

vectors

\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

Vector product yielding orthogonal vector

#vectors#cross-product

Vector Magnitude

vectors

|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Euclidean norm or length of a vector

#vectors#magnitude

Unit Vector

vectors

\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of length 1 pointing in same direction

#vectors#unit-vector

Scalar Projection of A onto B

vectors

\text{proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Magnitude of vector a in direction of vector b

#vectors#projection

Matrix Determinant (2x2)

vectors

\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

Determinant of a 2 by 2 matrix

#matrices#determinant

Matrix Inverse (2x2)

vectors

\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Inverse of an invertible 2 by 2 matrix

#matrices#inverse

Characteristic Equation (Eigenvalues)

vectors

\det(A - \lambda I) = 0

Equation for finding eigenvalues λ of matrix A

#matrices#eigenvalues

probability

Permutations Formula

probability

P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

probability

C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

probability

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

probability

P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

probability

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

probability

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

probability

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

probability

E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

probability

\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

probability

f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

trigonometry

Pythagorean Identity

Class 10trigonometry

\sin^2\theta + \cos^2\theta = 1

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry

1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry

1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry

\sin(2\theta) = 2\sin\theta\cos\theta

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry

\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#sines

Law of Cosines

Class 10trigonometry

c^2 = a^2 + b^2 - 2ab\cos C

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#cosines

logarithms

Product Rule for Logs

logarithms

\log_b(xy) = \log_b(x) + \log_b(y)

Logarithm of a product equals sum of logarithms

#product#logarithm

Quotient Rule for Logs

logarithms

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

Logarithm of a quotient equals difference of logarithms

#quotient#logarithm

Power Rule for Logs

logarithms

\log_b(x^k) = k \log_b(x)

Exponent comes out as a multiplier

#power#exponent#logarithm

Change of Base Formula

logarithms

\log_b(x) = \frac{\log_k(x)}{\log_k(b)} = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Converting logarithm to a different base k

#base#conversion

Identity & Zero Properties

logarithms

\log_b(b) = 1, \quad \log_b(1) = 0

Basic identity properties of logarithms

#identity#logarithm

Inverse Properties

logarithms

b^{\log_b x} = x, \quad \log_b(b^x) = x

Logarithms and exponentials are inverse functions

#inverse#exponential

Natural Log & Euler Constant

logarithms

\ln(x) = \log_e(x), \quad e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Definition of natural logarithm and base e

#natural#euler

Exponential Laws

logarithms

a^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

Fundamental laws of exponent arithmetic

#exponential#powers