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Product Rule for Indices

Topic: Indices and LogarithmsChapter: Chapter 1
am×an=am+na^m \times a^n = a^{m+n}
When multiplying exponential terms with the same base, add their exponents. 'a' is the base, and 'm', 'n' are exponents.

Quotient Rule for Indices

Topic: Indices and LogarithmsChapter: Chapter 1
aman=am−n\frac{a^m}{a^n} = a^{m-n}
When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator. 'a' is the base, and 'm', 'n' are exponents.

Power Rule for Indices

Topic: Indices and LogarithmsChapter: Chapter 1
(am)n=amn(a^m)^n = a^{mn}
When raising an exponential term to another power, multiply the exponents. 'a' is the base, and 'm', 'n' are exponents.

Power of a Product Rule

Topic: Indices and LogarithmsChapter: Chapter 1
(ab)m=ambm(ab)^m = a^m b^m
The power of a product is equal to the product of each factor raised to that power. 'a', 'b' are bases, and 'm' is the exponent.

Power of a Quotient Rule

Topic: Indices and LogarithmsChapter: Chapter 1
(ab)m=ambm\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}
The power of a quotient is equal to the quotient of the numerator and denominator each raised to that power. 'a', 'b' are bases, and 'm' is the exponent (b ≠ 0).

Zero Exponent Rule

Topic: Indices and LogarithmsChapter: Chapter 1
a0=1a^0 = 1
Any non-zero number raised to the power of zero is equal to 1 (a ≠ 0).

Negative Exponent Rule

Topic: Indices and LogarithmsChapter: Chapter 1
a−m=1ama^{-m} = \frac{1}{a^m}
A base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent. 'a' is the base, and 'm' is the exponent (a ≠ 0).

Fractional Exponent Rule (Root Form)

Topic: Indices and LogarithmsChapter: Chapter 1
am/n=amn=(an)ma^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m
A fractional exponent indicates both a power and a root. The numerator 'm' is the power, and the denominator 'n' is the root. 'a' is the base.

Definition of Logarithm

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = x
The logarithm of a number 'x' to a given base 'b' is the exponent 'y' to which 'b' must be raised to produce 'x'. 'b' is the base (b > 0, b ≠ 1), 'x' is the argument (x > 0).

Product Rule for Logarithms

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y
The logarithm of a product is the sum of the logarithms of the individual factors to the same base.

Quotient Rule for Logarithms

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y
The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator to the same base.

Power Rule for Logarithms

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡b(xn)=nlog⁡bx\log_b (x^n) = n \log_b x
The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number to the same base.

Change of Base Formula

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡bx=log⁡cxlog⁡cb\log_b x = \frac{\log_c x}{\log_c b}
This formula allows converting a logarithm from one base 'b' to another base 'c'. This is particularly useful for calculations with calculators, typically using base 10 or base 'e'.

Logarithm of Base to Itself

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡bb=1\log_b b = 1
The logarithm of a number to its own base is always 1.

Logarithm of One

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡b1=0\log_b 1 = 0
The logarithm of 1 to any valid base is always 0.

Inverse Property of Logarithms

Topic: Indices and LogarithmsChapter: Chapter 1
blog⁡bx=xb^{\log_b x} = x
Raising the base 'b' to the power of the logarithm of 'x' to base 'b' results in 'x'. This highlights the inverse relationship between exponentiation and logarithms.

Reciprocal Property of Logarithms

Topic: Indices and LogarithmsChapter: Chapter 1
log⁡ba=1log⁡ab\log_b a = \frac{1}{\log_a b}
The logarithm of 'a' to base 'b' is the reciprocal of the logarithm of 'b' to base 'a'.

Ratio Definition

Topic: Logical ReasoningChapter: Chapter 1
a:b or aba:b \text{ or } \frac{a}{b}
A ratio expresses the relationship between two quantities of the same kind. Here, 'a' is the antecedent and 'b' is the consequent.

Compound Ratio

Topic: Logical ReasoningChapter: Chapter 1
ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
The compound ratio of two or more ratios is obtained by multiplying their antecedents and consequents separately.

Duplicate Ratio

Topic: Logical ReasoningChapter: Chapter 1
a2:b2a^2 : b^2
The duplicate ratio of a given ratio a:b is a squared to b squared.

Sub-duplicate Ratio

Topic: Logical ReasoningChapter: Chapter 1
a:b\sqrt{a} : \sqrt{b}
The sub-duplicate ratio of a given ratio a:b is the square root of a to the square root of b.

Triplicate Ratio

Topic: Logical ReasoningChapter: Chapter 1
a3:b3a^3 : b^3
The triplicate ratio of a given ratio a:b is a cubed to b cubed.

Sub-triplicate Ratio

Topic: Logical ReasoningChapter: Chapter 1
a1/3:b1/3a^{1/3} : b^{1/3}
The sub-triplicate ratio of a given ratio a:b is the cube root of a to the cube root of b.

Inverse Ratio (Reciprocal Ratio)

Topic: Logical ReasoningChapter: Chapter 1
b:a or bab:a \text{ or } \frac{b}{a}
The inverse ratio of a given ratio a:b is b:a.

Proportion Definition

Topic: Logical ReasoningChapter: Chapter 1
a:b::c:d or ab=cda:b :: c:d \text{ or } \frac{a}{b} = \frac{c}{d}
A proportion states that two ratios are equal. 'a' and 'd' are extremes, 'b' and 'c' are means.

Product of Extremes and Means

Topic: Logical ReasoningChapter: Chapter 1
If ab=cd, then ad=bc\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } ad = bc
In a proportion, the product of the extremes is equal to the product of the means.

Continued Proportion

Topic: Logical ReasoningChapter: Chapter 1
If a:b::b:c, then b2=ac\text{If } a:b :: b:c, \text{ then } b^2 = ac
Three quantities a, b, c are in continued proportion if a:b = b:c. Here, 'b' is the mean proportional.

Componendo Rule

Topic: Logical ReasoningChapter: Chapter 1
If ab=cd, then a+bb=c+dd\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } \frac{a+b}{b} = \frac{c+d}{d}
If two ratios are equal, then the ratio of the sum of the antecedent and consequent to the consequent is also equal.

Dividendo Rule

Topic: Logical ReasoningChapter: Chapter 1
If ab=cd, then a−bb=c−dd\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } \frac{a-b}{b} = \frac{c-d}{d}
If two ratios are equal, then the ratio of the difference of the antecedent and consequent to the consequent is also equal.

Componendo and Dividendo Rule

Topic: Logical ReasoningChapter: Chapter 1
If ab=cd, then a+ba−b=c+dc−d\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } \frac{a+b}{a-b} = \frac{c+d}{c-d}
If two ratios are equal, then the ratio of the sum of antecedent and consequent to their difference is also equal.

Product Rule of Indices

Topic: Logical ReasoningChapter: Chapter 1
am×an=am+na^m \times a^n = a^{m+n}
When multiplying exponential terms with the same base, add the exponents.

Quotient Rule of Indices

Topic: Logical ReasoningChapter: Chapter 1
aman=am−n\frac{a^m}{a^n} = a^{m-n}
When dividing exponential terms with the same base, subtract the exponents.

Power Rule of Indices

Topic: Logical ReasoningChapter: Chapter 1
(am)n=amn(a^m)^n = a^{mn}
When raising an exponential term to another power, multiply the exponents.

Zero Exponent Rule

Topic: Logical ReasoningChapter: Chapter 1
a0=1(aeq0)a^0 = 1 \quad (a eq 0)
Any non-zero number raised to the power of zero is 1.

Negative Exponent Rule

Topic: Logical ReasoningChapter: Chapter 1
a−m=1ama^{-m} = \frac{1}{a^m}
A base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent.

Fractional Exponent Rule

Topic: Logical ReasoningChapter: Chapter 1
am/n=(an)m=amna^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}
A fractional exponent indicates both a root and a power. The denominator 'n' is the root, and the numerator 'm' is the power.

Product to Power Rule

Topic: Logical ReasoningChapter: Chapter 1
(ab)m=ambm(ab)^m = a^m b^m
When a product of bases is raised to an exponent, each base is raised to that exponent.

Quotient to Power Rule

Topic: Logical ReasoningChapter: Chapter 1
(ab)m=ambm(\frac{a}{b})^m = \frac{a^m}{b^m}
When a quotient of bases is raised to an exponent, both the numerator and denominator are raised to that exponent.

Logarithm Definition

Topic: Logical ReasoningChapter: Chapter 1
If ax=N, then log⁡aN=x(a>0,aeq1,N>0)\text{If } a^x = N, \text{ then } \log_a N = x \quad (a>0, a eq 1, N>0)
The logarithm of a number N to a given base 'a' is the exponent 'x' to which the base must be raised to produce N.

Product Rule of Logarithms

Topic: Logical ReasoningChapter: Chapter 1
log⁡b(MN)=log⁡bM+log⁡bN\log_b (MN) = \log_b M + \log_b N
The logarithm of a product of two numbers is the sum of their logarithms.

Quotient Rule of Logarithms

Topic: Logical ReasoningChapter: Chapter 1
log⁡b(MN)=log⁡bM−log⁡bN\log_b (\frac{M}{N}) = \log_b M - \log_b N
The logarithm of a quotient of two numbers is the difference of their logarithms.

Power Rule of Logarithms

Topic: Logical ReasoningChapter: Chapter 1
log⁡b(Mp)=p⋅log⁡bM\log_b (M^p) = p \cdot \log_b M
The logarithm of a number raised to a power is the product of the power and the logarithm of the number.

Change of Base Rule

Topic: Logical ReasoningChapter: Chapter 1
log⁡bM=log⁡cMlog⁡cb\log_b M = \frac{\log_c M}{\log_c b}
Allows converting a logarithm from one base 'b' to another base 'c'.

Logarithm of 1

Topic: Logical ReasoningChapter: Chapter 1
log⁡b1=0\log_b 1 = 0
The logarithm of 1 to any valid base is always 0.

Logarithm of Base

Topic: Logical ReasoningChapter: Chapter 1
log⁡bb=1\log_b b = 1
The logarithm of a number to the same base is always 1.

Inverse Property of Logarithms

Topic: Logical ReasoningChapter: Chapter 1
blog⁡bM=Mb^{\log_b M} = M
The exponential function and logarithmic function with the same base are inverse operations.

Reciprocal Property of Logarithms

Topic: Logical ReasoningChapter: Chapter 1
log⁡ba=1log⁡ab\log_b a = \frac{1}{\log_a b}
The logarithm of 'a' to base 'b' is the reciprocal of the logarithm of 'b' to base 'a'.

Simple Interest (SI)

Topic: Time Value of MoneyChapter: Chapter 1
SI=P×R×T100SI = \frac{P \times R \times T}{100}
Calculates interest only on the principal amount. P is Principal, R is Rate of Interest per annum (in percentage), T is Time period in years.

Amount (A) for Simple Interest

Topic: Time Value of MoneyChapter: Chapter 1
A=P+SI=P(1+R×T100)A = P + SI = P \left(1 + \frac{R \times T}{100}\right)
The total amount payable at the end of the period, including principal and simple interest. A is Amount, P is Principal, R is Rate of Interest per annum (in percentage), T is Time period in years.

Amount (A) for Compound Interest (Future Value)

Topic: Time Value of MoneyChapter: Chapter 1
A=P(1+im)n×mA = P \left(1 + \frac{i}{m}\right)^{n \times m}
Calculates the future value of an investment where interest is compounded periodically. A is Amount (Future Value), P is Principal (Present Value), i is Annual Nominal Interest Rate (as a decimal), m is Number of compounding periods per year, n is Number of years. When compounding annually, m=1, so A = P(1+i)^n. This formula heavily uses indices.

Present Value (PV) for Compound Interest

Topic: Time Value of MoneyChapter: Chapter 1
PV=A(1+im)n×m=A(1+im)−n×mPV = \frac{A}{\left(1 + \frac{i}{m}\right)^{n \times m}} = A \left(1 + \frac{i}{m}\right)^{-n \times m}
Calculates the current value of a future sum of money, discounted at a specific compound interest rate. PV is Present Value, A is Future Value, i is Annual Nominal Interest Rate (as a decimal), m is Number of compounding periods per year, n is Number of years.

Effective Annual Rate (EAR)

Topic: Time Value of MoneyChapter: Chapter 1
EAR=(1+im)m−1EAR = \left(1 + \frac{i}{m}\right)^m - 1
Represents the actual annual rate of interest earned or paid, considering the effect of compounding more frequently than annually. i is Annual Nominal Interest Rate (as a decimal), m is Number of compounding periods per year.

Future Value of an Ordinary Annuity (FVOA)

Topic: Time Value of MoneyChapter: Chapter 1
FVOA=Pmt[(1+i)n−1i]FVOA = Pmt \left[ \frac{\left(1 + i\right)^n - 1}{i} \right]
Calculates the future value of a series of equal payments (Pmt) made at the end of each period, compounded at a rate i for n periods. Pmt is Payment per period, i is Interest rate per period (as a decimal), n is Number of periods.

Present Value of an Ordinary Annuity (PVOA)

Topic: Time Value of MoneyChapter: Chapter 1
PVOA=Pmt[1−(1+i)−ni]PVOA = Pmt \left[ \frac{1 - \left(1 + i\right)^{-n}}{i} \right]
Calculates the present value of a series of equal payments (Pmt) made at the end of each period, discounted at a rate i for n periods. Pmt is Payment per period, i is Interest rate per period (as a decimal), n is Number of periods.

Future Value of an Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 1
FVAD=Pmt[(1+i)n−1i](1+i)FVAD = Pmt \left[ \frac{\left(1 + i\right)^n - 1}{i} \right] \left(1 + i\right)
Calculates the future value of a series of equal payments (Pmt) made at the beginning of each period, compounded at a rate i for n periods. It is the FVOA multiplied by (1+i).

Present Value of an Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 1
PVAD=Pmt[1−(1+i)−ni](1+i)PVAD = Pmt \left[ \frac{1 - \left(1 + i\right)^{-n}}{i} \right] \left(1 + i\right)
Calculates the present value of a series of equal payments (Pmt) made at the beginning of each period, discounted at a rate i for n periods. It is the PVOA multiplied by (1+i).

Ratio Definition

Topic: Ratio and ProportionChapter: Chapter 1
Ifaandbaretwoquantitiesofthesamekind(beq0),theirratioisexpressedasa:borab.If a and b are two quantities of the same kind (b eq 0), their ratio is expressed as a:b or \frac{a}{b}.
A ratio compares two quantities of the same unit. It indicates how many times one quantity contains the other. The quantities 'a' and 'b' are called terms of the ratio, where 'a' is the antecedent and 'b' is the consequent.

Inverse Ratio

Topic: Ratio and ProportionChapter: Chapter 1
Theinverseratioofa:bisb:aor1a:1b.The inverse ratio of a:b is b:a or \frac{1}{a}:\frac{1}{b}.
The inverse ratio is obtained by interchanging the antecedent and the consequent.

Compound Ratio

Topic: Ratio and ProportionChapter: Chapter 1
Thecompoundratioofa:bandc:dis(a×c):(b×d)ora×cb×d.The compound ratio of a:b and c:d is (a \times c) : (b \times d) or \frac{a \times c}{b \times d}.
A compound ratio is obtained by multiplying the antecedents together and the consequents together of two or more ratios.

Duplicate Ratio

Topic: Ratio and ProportionChapter: Chapter 1
Theduplicateratioofa:bisa2:b2.The duplicate ratio of a:b is a^2:b^2.
It's the ratio of the squares of the terms of a given ratio.

Sub-duplicate Ratio

Topic: Ratio and ProportionChapter: Chapter 1
Thesub−duplicateratioofa:bisa:b.The sub-duplicate ratio of a:b is \sqrt{a}:\sqrt{b}.
It's the ratio of the square roots of the terms of a given ratio.

Triplicate Ratio

Topic: Ratio and ProportionChapter: Chapter 1
Thetriplicateratioofa:bisa3:b3.The triplicate ratio of a:b is a^3:b^3.
It's the ratio of the cubes of the terms of a given ratio.

Sub-triplicate Ratio

Topic: Ratio and ProportionChapter: Chapter 1
Thesub−triplicateratioofa:bisa3:b3.The sub-triplicate ratio of a:b is \sqrt[3]{a}:\sqrt[3]{b}.
It's the ratio of the cube roots of the terms of a given ratio.

Proportion Definition

Topic: Ratio and ProportionChapter: Chapter 1
Fourquantitiesa,b,c,dareinproportionifa:b=c:d,whichcanbewrittenasab=cd.Here,aanddareextremes,andbandcaremeans.Theproductofextremesequalstheproductofmeans:ad=bc.Four quantities a, b, c, d are in proportion if a:b = c:d, which can be written as \frac{a}{b} = \frac{c}{d}. Here, a and d are extremes, and b and c are means. The product of extremes equals the product of means: ad = bc.
Proportion signifies the equality of two ratios. This fundamental property (product of extremes = product of means) is key to solving proportion problems.

Continued Proportion

Topic: Ratio and ProportionChapter: Chapter 1
Threequantitiesa,b,careincontinuedproportionifa:b=b:c,whichmeansb2=ac.Here,biscalledthemeanproportionalbetweenaandc.Three quantities a, b, c are in continued proportion if a:b = b:c, which means b^2 = ac. Here, b is called the mean proportional between a and c.
In continued proportion, the consequent of the first ratio is the antecedent of the second ratio. This leads to the relationship where the square of the middle term equals the product of the other two.

Mean Proportional

Topic: Ratio and ProportionChapter: Chapter 1
Themeanproportionalbetweenaandcisac.The mean proportional between a and c is \sqrt{ac}.
Derived from continued proportion, the mean proportional is the square root of the product of the two given quantities.

Invertendo Property of Proportion

Topic: Ratio and ProportionChapter: Chapter 1
Ifa:b=c:d,thenb:a=d:corba=dc.If a:b = c:d, then b:a = d:c or \frac{b}{a} = \frac{d}{c}.
If two ratios are equal, then their inverse ratios are also equal. This property allows for inversion of both sides of a proportion.

Alternendo Property of Proportion

Topic: Ratio and ProportionChapter: Chapter 1
Ifa:b=c:d,thena:c=b:dorac=bd.If a:b = c:d, then a:c = b:d or \frac{a}{c} = \frac{b}{d}.
If two ratios are equal, then the ratio of the first antecedent to the second antecedent is equal to the ratio of the first consequent to the second consequent. This property allows for swapping of means.

Componendo Property of Proportion

Topic: Ratio and ProportionChapter: Chapter 1
Ifa:b=c:d,then(a+b):b=(c+d):dora+bb=c+dd.If a:b = c:d, then (a+b):b = (c+d):d or \frac{a+b}{b} = \frac{c+d}{d}.
If two ratios are equal, then the ratio of the sum of the terms of the first ratio to its consequent is equal to the ratio of the sum of the terms of the second ratio to its consequent.

Dividendo Property of Proportion

Topic: Ratio and ProportionChapter: Chapter 1
Ifa:b=c:d,then(a−b):b=(c−d):dora−bb=c−dd.If a:b = c:d, then (a-b):b = (c-d):d or \frac{a-b}{b} = \frac{c-d}{d}.
If two ratios are equal, then the ratio of the difference of the terms of the first ratio to its consequent is equal to the ratio of the difference of the terms of the second ratio to its consequent.

Componendo and Dividendo Property of Proportion

Topic: Ratio and ProportionChapter: Chapter 1
Ifa:b=c:d,then(a+b):(a−b)=(c+d):(c−d)ora+ba−b=c+dc−d.If a:b = c:d, then (a+b):(a-b) = (c+d):(c-d) or \frac{a+b}{a-b} = \frac{c+d}{c-d}.
This property combines Componendo and Dividendo. It is extremely useful in simplifying complex proportional equations and is frequently tested.

Addendo Property of Proportion

Topic: Ratio and ProportionChapter: Chapter 1
Ifab=cd=ef=...,theneachratioisequaltoa+c+e+...b+d+f+....If \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = ..., then each ratio is equal to \frac{a+c+e+...}{b+d+f+...}.
If several ratios are equal, then each individual ratio is also equal to the ratio of the sum of all antecedents to the sum of all consequents.
Chapter 1 Formulas — CA Foundation | Kartavvya