Ratio, Proportion, Indices, LogarithmMCQPYQ Jan 26Question 4548 of 305
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If 3x×b5log⁡bx=192\displaystyle 3x \times b^5 \log_b x = 192 then the value of x is

Options

A8
B4
C2
D22\displaystyle 2\sqrt{2}
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Correct Answer

✅ Option c — 2

All Options:

  • A8
  • B4
  • C2
  • D22\displaystyle 2\sqrt{2}

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Detailed Solution & Explanation

Given equation:
3x×b5log⁡bx=1923x \times b^{5 \log_b x} = 192

Using the logarithmic property nlog⁡am=log⁡a(mn)\displaystyle n \log_a m = \log_a (m^n):
5log⁡bx=log⁡b(x5)5 \log_b x = \log_b (x^5)b5log⁡bx=blog⁡b(x5)b^{5 \log_b x} = b^{\log_b (x^5)}

Using the identity blog⁡by=y\displaystyle b^{\log_b y} = y:
blog⁡b(x5)=x5b^{\log_b (x^5)} = x^5

Now substitute this back into the original equation:
3x×x5=1923x \times x^5 = 192
3x6=1923x^6 = 192
Divide both sides by 3\displaystyle 3:
x6=1923x^6 = \frac{192}{3}
x6=64x^6 = 64

Express 64\displaystyle 64 as a power of 2\displaystyle 2:
64=2664 = 2^6
x6=26  ⟹  x=2x^6 = 2^6 \implies x = 2

Hence, **Option C** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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