Question

If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4), then a equals

3

Solution

From CAT 2017 – Quantitative Aptitude – Algebra – Logarithms, we can see that,
log (2^a. 3^b. 5^c) = [log (2^2.3^3.5) + log (2^6.3.5^7) + log (2.3^2.5^4)]/3
3 * log (2^a. 3^b. 5^c) = log (2^9.3^6.5^12)
log (2^a. 3^b. 5^c)^3 = log (2^9.3^6.5^12)
log (2^3a. 3^3b. 5^3c) = log (2^9.3^6.5^12)
3a = 9
a=3

Logarithm Concepts Questions and Answers for CAT 2018 Quant Preparation

Q1: If x is a real number such that log(base 3)5 = log(base 5)(2 + x), then which of the following is true?

Q2: The value of log (base 0.008) √5 + log (base√3) 81 – 7 is equal to

Q3: Suppose, log(base3)x = log(base12)y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log(base6)G is equal to

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Question

If x is a real number such that log(base 3)5 = log(base 5)(2 + x), then which of the following is true?

A) 0 < x < 3
B) 23 < x < 30
C) x > 30
D) 3 < x < 23

Option (D)

Solution

From CAT 2017 – Quantitative Aptitude – Algebra – Logarithms, we can see that,
Log(base 3)5 lies between 1 and 2 because Log(base 3)3 = 1 and Log(base 3)9 = 2
1 < Log(base 3)5 < 2
So, log(base 5)(2+x) should also lie between 1 and 2
1 < log(base 5)(2+x) < 2
5^1 < 2+x < 5^2
5 < 2+x < 25
3 < x < 23
Option D is the right answer.

Logarithm Concepts Questions and Answers for CAT 2018 Quant Preparation

Q1: If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4), then a equals

Q2: The value of log (base 0.008) √5 + log (base√3) 81 – 7 is equal to

Q3: Suppose, log(base3)x = log(base12)y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log(base6)G is equal to

Online Coaching Course for CAT Exam Preparation

a) 750+ Videos covering entire CAT syllabus
b) 2 Live Classes (online) every week for doubt clarification
c) Study Material & PDFs for practice and understanding
d) 10 Mock Tests in the latest pattern
e) Previous Year Questions solved on video

Question

The value of log (base 0.008) √5 + log (base√3) 81 – 7 is equal to

A) 1/3
B) 2/3
C) 5/6
D) 7/6

Option (C)

Solution

As per CAT 2017 – Quantitative Aptitude – Algebra – Logarithms, we can see that
log(base 0.008)5^(1/2) = -1/6
Log (base 3^1/2) 3^4 = 8
So, -1/6 + 8 – 7 = 5/6
Option (C)

Logarithm Concepts Questions and Answers for CAT 2018 Quant Preparation

Q1: If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4), then a equals

Q2: If x is a real number such that log(base 3)5 = log(base 5)(2 + x), then which of the following is true?

Q3: Suppose, log(base3)x = log(base12)y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log(base6)G is equal to

Online Coaching Course for CAT Exam Preparation

a) 750+ Videos covering entire CAT syllabus
b) 2 Live Classes (online) every week for doubt clarification
c) Study Material & PDFs for practice and understanding
d) 10 Mock Tests in the latest pattern
e) Previous Year Questions solved on video

Question

Suppose, log(base3)x = log(base12)y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log(base6)G is equal to

A) √a
B) 2a
C) a/2
D) a

Option (D)

Solution

As per CAT 2017 – Quantitative Aptitude – Algebra – Logarithms, we can see that
x=3^a and y=12^a
G = √(3^a * 12^a) = 6^a
Log (base 6) 6^a = a
Option (D)

Logarithm Concepts Questions and Answers for CAT 2018 Quant Preparation

Q1: If log (2^a × 3^b × 5^c) is the arithmetic mean of log (2^2 × 3^3 × 5), log (2^6 × 3 × 5^7), and log(2 × 3^2 × 5^4), then a equals

Q2: If x is a real number such that log(base 3)5 = log(base 5)(2 + x), then which of the following is true?

Q3: The value of log (base 0.008) √5 + log (base√3) 81 – 7 is equal to