Logarithm Concepts for CAT

At least one question on the CAT has to come from this part. Whenever there is a question about logarithm, the question is about how logarithm is used, and the reasoning behind the question is the use of logarithm. Another thing I want to say is that many students are afraid of this chapter. But I think they should not worry about the word "logarithm." Instead, they should think of logarithm problems as problems with exponents, and most of them can be answered quickly by plotting a graph. This part is important for tests like SNAP, MAT, IIFT, JMET, and XAT, among others.

Table of Content

Exponential Function
Logarithm
Types of Logarithms
Logarithm properties
Characteristics and Mantissa
Points to Remember About Characteristics
Important Conversions
Previous year CAT questions
How to approach Logarithm questions in CAT

Exponential Function

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For every x ∈ R, e^x = \(1+ x+ \frac{x^2}{2!} + \frac{x^3}{3!} + ......or\ e^x = \Sigma \frac{x^n}{n!}\), where n = 1 to ∞.

e^x is exponential function and it is a finite number for every x ∈R.

graph of ex

Properties

For every x ∈ R, e^x is defined, then

(i) e^x > 0 for all x ∈R and e⁰ =1

(ii) e^a >e^b if a>b and a,b∈R

(iii) e^a × e^b = e^{(a + b)} for all a, b∈R

(iv) e^a ÷ e^b =e^{(a − b)} for all a, b∈R

(v) (e^a)^b = e^ab for all a, b∈R

(vi) For each positive real number x there exists one and only one real number y such that e^y = x

(vii) e^x is one-one function

(viii) e ≈ 2.714

Logarithm

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Let a,b be positive real numbers

So, a^x =b can be written as

log_a b = x a \(\neq 1\), a > 0, b > 0

Ex:- 3⁵ = 243 \(\Leftrightarrow\) log₃ 243 = 5

Types of Logarithms

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(I) Natural Logarithm : log_e N is called natural Logarithm or Naperian Logarithm denoted by (ln N) i.e., when logarithm’s base is “e” then it is called natural logarithm. Ex: log_e 7

(II) Common Logarithm : log₁₀ N is called Brigg’s Logarithm when the base is 10. Ex: log₁₀100

Logarithm properties

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log_a 1 = 0, a > 0, a \(\neq\) 1
log_a a = 1, a > 0, a \(\neq\) 1
Log_a a^x = x, x \(\forall\) x \(\in\) R, x >0
a^{log_a x} = x, x \(\forall\) x \(\in\) R, x >0
log_a (m.n) = log_a m + log_a n, n \(\forall\) m, n > 0, a > 0, a \(\neq\) 1
log_a (m/n) = log_a m – log_a n, n \(\forall\) m, n > 0, a > 0, a \(\neq\) 1
log_a (mⁿ) = n log_a m, m \(\forall\) m, m > 0, a > 0, a \(\neq\) 1
log_a (\(\frac{1}{m}\)) = – log_a m, m \(\forall\) m, n > 0, a > 0, a \(\neq\) 1
log_a (b) = \(\frac{1}{log_b a} = \frac{log_C b}{log_c a}\)\(\forall\) a,b,c > 0, a \(\neq\)1, b \(\neq\)1, c \(\neq\)1
log_a b = x \(\forall\) a, b > 0, a \(\neq\) 1 and x \(\in\) R

(I) log_1/a b = – x (II) log_a \((\frac{1}{b})\) = – x (III) log_1/a \((\frac{1}{b})\) = – x

log_a^m b = \(\frac{1}{m}\) log_a b

log_a x is a decreasing function, if 0 < a < 1

log_a x is a increasing function, if a > 1

When 0 < a < 1 then

log_a b \(\geq\) log_a c, \(\Leftrightarrow\) b \( \leq\) c

log_a b \(\geq\) c, \(\Leftrightarrow\) b \( \leq\) a^c

When a >1

log_a b \(\geq\) log_a c, \(\Leftrightarrow\) b \(\geq\) c > 0

log_a b \(\geq\) c, \(\Leftrightarrow\) b \(\geq\) a^c

Characteristics and Mantissa

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Characteristic: The integral part of logarithm is known as characteristic.

Mantissa: The decimal part is known as mantissa and is always positive

In log 3274 = 3.5150, the integral part is 3 i.e., characteristic is 3 and the decimal part is .5150 i.e., mantissa is .5150.

Points to Remember About Characteristics

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The characteristic of common logarithm of positive number less than unity (i.e.,1) is negative.
The characteristic of common logarithm of a positive number greater than 1 is positive.
If the logarithm to any base a gives the characteristic ‘n’, then the number of possible integral values is given by aⁿ⁺¹ −aⁿ. For example log₁₀ x = n.abcd, then the number of integral values that x can have given by 10ⁿ⁺¹ −10ⁿ

If the characteristic of log₁₀ x is negative (i.e., − n), then the number of zeros between the decimal and the first significant number after the decimal is (n −1)

Important Conversions

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For a > 1, a 1, c > 0, a ^b > c \(\Leftrightarrow\) log_a c < b
For 0 < a < 1, a 1, c > 0, a ^b > c \(\Leftrightarrow\) log_a c > b

Previous year CAT questions

Ques 1: Find x, if \(log_{2x} \sqrt x + log_{2\sqrt x} x = 0\). (CAT 2015)

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Solution:

calulations

log₂x = 0 then x = 2⁰=1

log₂x = -(6/5) then x = 2^-(6/5)

x = 1 and x = 2^-(6/5)

Ques 2: If x ≥ y and y > 1, then the value of the expression log_x (\(\frac{x}{y}\)) + log_y (\(\frac{y}{x}\))can never be (CAT 2014)

−1
−0.5
0
1

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Solution: (D)

Let P = log_x \(\frac{x}{y}\)+ log_y\(\frac{y}{x}\)

= log_xx - log_xy + log_yy - log_yx

=2 - log_xy – log_yx

Again let t = log_xy

p = 2 – \(\frac{1}{t}\) – t = – \((\sqrt t-\frac{1}{\sqrt t}) ^2\)

Which can never be 1.

Ques 3: What is the sum of n terms in the series log m + log \((\frac{m^2}{n})\) + log \((\frac{m^3}{n^2})\) + log \((\frac{m^4}{n^3})\) …….? (CAT 2014)

\(log [ \frac{n^{(n-1)}}{m^{(n+1)}} ]^{n/2}\)
\(log [ \frac{m^m}{n^n} ]^{n/2}\)
\(log [ \frac{m^ {(1-n)}}{n^{(1-m)}} ]^{n/2}\)
\(log [ \frac{m^{(n+1)}}{n^{(n-1)}} ]^{n/2}\)

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Solution:- (D)

calculation

Ques 4: Find the values of x and y for the given equations (CAT 2013)

xy²= 4
log₃ (log₂ x) + log_1/3 (log_1/2 x) = 1

X= \(\frac{1}{8}\) , y =64
x= 64, y = \(\frac{1}{4}\)
x= 16 , y = \(\frac{1}{2}\)
x = \(\frac{1}{16}\), y= 48

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Solution:- (B)

log₃ (log₂ x)+log _1/3 (log _½ y)=1

log₃(log₂x)-log₃(log _½ y) = 1

calculation

x=(1/y³), xy³=1

xy²=4

From Equation (i) and (ii) we get x= 64, y =(1/4)

Ques 5: log_x (a - b) – log_x (a + b) = \(log_x (\frac{b}{a})\) Find \(\frac{a^2}{b^2} + \frac{b ^2}{a^2}\) . (CAT 2012)

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Solution: (D)

log_xa− b/a + b = log_xb/a

⇒ a(a−b) = b(a+b)

⇒ a²−ab = ab+b²

⇒ a²−b²= 2ab

⇒ a²-2ab – b²= 0

⇒ \((\frac{a}{b})^2- 2 (\frac{a}{b}) -1 + 0\)

This is a quadratic in a/b. The product of the roots is −1,

i.e., if a is a root, then \((\frac{-1}{\alpha})\) will also be root i.e.,

if \(\frac{a}{b}\) (or \(\alpha\)) is one root, then the other root is \(\frac{-b}{a}\) .

\((\frac{a}{b} ) + ( \frac{b}{a}) = \alpha ^2 + \frac{1}{\alpha ^2}\)

\((\alpha + \frac{-1}{\alpha})^2 + 2 = 2^2 + 2 = 6\)

Ques 6: Which of the following statements is not correct? (CAT 2010)

log₁₀10 = 1
log (2+3) = log (2×3)
log₁₀1 = 0
log (1+2+3) log1 + log2 + log3

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Solution: (B)

From the property of logarithms. log(a × b) = log a + log b

Ques 7: Let u = (log₂ x)²− 6log₂ x + 12, where x is a real number. Then, the equation x^u = 256, has __ . (CAT 2010)

no solution for x
exactly one solution for x
exactly two distinct solutions for x
exactly three distinct solutions for x

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Solution: (B)

x^u =256

u log₂ x = 8 (2⁸=256)

Let log x = p

P³ − 6p² +12p−8=0

p²(p−2)−4p(p−2)+4(p−2)=0

(p² −4p+4)(p−2)=0

(p−2)³ =0

p = 2

Hence, exactly one solution.

Ques 8: If then which of the following pairs of values of (a,b) is not possible? (CAT 2006)

(-2,1/2)
(1,1)
(0.4, 2.5)
(\(\pi, \frac{1}{\pi}\))
(2,2)

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Solution: (E)

calculation

Ques 9: If x ≥ y and y > 1, then the value of the expression log_x (\(\frac{x}{y}\)) + log_y (\(\frac{y}{x}\)) can never be. (CAT 2014)

−1
−0.5
0
1

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Solution: (D)

Let P = log_x \(\frac{x}{y}\)+ log_y\(\frac{y}{x}\)

= log_xx - log_xy + log_yy - log_yx

=2 - log_xy – log_yx

Again let t = log_xy

p = 2 – \(\frac{1}{t}\) – t = – \((\sqrt t-\frac{1}{\sqrt t}) ^2\)

Ques 10: Let u = (log₂ x)²− 6log₂ x + 12, where x is a real number. Then, the equation x^u = 256, has __ . (CAT 2010)

no solution for x
exactly one solution for x
exactly two distinct solutions for x
exactly three distinct solutions for x

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Solution: (B)

u = (log₂x)²− 6(log₂x) + 12

Let log₂ x = p ...(i)

⇒ u = p² − 6 p + 12

x^u =256(=2⁸)

Applying log to base 2 on both sides, we get

ulog₂x=log₂2⁸,

ulog₂x=8 ...(ii)

Dividing Eq. (ii) by Eq, (i), we get

u = 8 / p ⇒ 8 / p = p2 − 6 p + 12

⇒ 8−p³ −6p² +12p

⇒ p³ −6p² +12p−8 = 0

⇒ (p−2)³ = 0

⇒ p = 2

log₂x = 2

⇒x=2² = 4

So, we have exactly one solution.

Ques 11: If log₃ 2, log₃(2^x − 5), log₃(2^x − 7 / 2) are in arithmetic progression, then the value of x is equal to (CAT 2003)

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Solution:- (D)

the three terms in AP are a, b, c which are related as 2b = a + c

2 [log₃( 2^x – 5)] = log₃ 2 + log₃(2x- \(-\frac{7}{2}\))

log(2x-5)²=(2^(x+1)- 7)

Putting the options provided , only x = 3 satisfies the condition.

Ques 12: If \(\frac{1}{3}\) log₃ M + 3 log₃ N = 1 + log_0.0085, then (CAT 2003)

M⁹ =(9/N)
N⁹ = (9/M)
M³=(3/N)
N⁹=(3/M)

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Solution: (B)

calculation

Ques 13: If log₁₀ x − log₁₀\(\sqrt x\) = 2 log_x 10, then a possible value of x is given by (CAT 2003)

10
\(\frac{1}{100}\)
\(\frac{1}{1000}\)
None of these

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Solution: (B)

calculation

Ques 14: log₂[ log₇(x² − x + 37) ] = 1, then what could be the value of x ? (CAT 1997)

3
5
4
None of these

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Solution: (C)

log₂[log₇ (x² − x + 37)] = 1

using log _p x = y

⇒p^y = x

∴ 2 = log₇ (x² − x + 37)

⇒ 49 = x² − x + 37

⇒ x² − x − 12 = 0 ⇒ (x − 4)(x + 3) = 0

∴ x=4, -3

So the solution is 4 based on the options given.

Ques 15: If log ₇ log₅\((\sqrt{ x+5} + \sqrt x) = 0\) . Find the value of x. (CAT 1994)

1
0
2
None of these

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Solution: (B)

calculation

How to approach Logarithm questions in CAT

Formulae and properties are crucial for attemting questions in CAT from this chapter
Go through every formulae and property and the interchangeable forms for easy calculation.
Try to attempt questions from series and logarithms for preparing for CAT.

Logarithm Concepts for CAT

Exponential Function

Properties

Logarithm

Types of Logarithms

Logarithm properties

Characteristics and Mantissa

Points to Remember About Characteristics

Important Conversions

Previous year CAT questions

How to approach Logarithm questions in CAT

CAT Related Questions

1.
The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:

2.
The set of all real values of $x$ for which $(x^2 - |x+9| + x)>0$ is:

3.
The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:

4.
Let $a_n$ be the $n^{\text{th}}$ term of a decreasing infinite geometric progression. If $a_1 + a_2 + a_3 = 52$ and $a_1a_2 + a_2a_3 + a_3a_1 = 624$, then the sum of this geometric progression is:

5.
A loan of Rs 1000 is fully repaid by two installments of Rs 530 and Rs 594, paid at the end of the first and second year, respectively. If the interest is compounded annually, then the rate of interest, in percentage, is:

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Logarithm Concepts for CAT

Exponential Function

Properties

Logarithm

Types of Logarithms

Logarithm properties

Characteristics and Mantissa

Points to Remember About Characteristics

Important Conversions

Previous year CAT questions

How to approach Logarithm questions in CAT

CAT Related Questions

1.The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:

2.The set of all real values of $x$ for which $(x^2 - |x+9| + x)>0$ is:

3.The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:

4.Let $a_n$ be the $n^{\text{th}}$ term of a decreasing infinite geometric progression. If $a_1 + a_2 + a_3 = 52$ and $a_1a_2 + a_2a_3 + a_3a_1 = 624$, then the sum of this geometric progression is:

5.A loan of Rs 1000 is fully repaid by two installments of Rs 530 and Rs 594, paid at the end of the first and second year, respectively. If the interest is compounded annually, then the rate of interest, in percentage, is:

Comments

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1.
The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:

2.
The set of all real values of $x$ for which $(x^2 - |x+9| + x)>0$ is:

3.
The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:

4.
Let $a_n$ be the $n^{\text{th}}$ term of a decreasing infinite geometric progression. If $a_1 + a_2 + a_3 = 52$ and $a_1a_2 + a_2a_3 + a_3a_1 = 624$, then the sum of this geometric progression is:

5.
A loan of Rs 1000 is fully repaid by two installments of Rs 530 and Rs 594, paid at the end of the first and second year, respectively. If the interest is compounded annually, then the rate of interest, in percentage, is: