Ratio, Proportion, Indices, LogarithmMCQPYQ July 21Question 938 of 305
All Questions

If log⁡x+log⁡16+log⁡x+log⁡256=256\displaystyle \log x + \log 16 + \log x + \log 256 = \frac{25}{6} then the value of x\displaystyle x is

Options

A64
B4
C16
D2
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Correct Answer

✅ Option c — 16

All Options:

  • A64
  • B4
  • C16
  • D2

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Detailed Solution & Explanation

Let us analyze the given equation. As written in the question text:
log⁡x+log⁡16+log⁡x+log⁡256=256\log x + \log 16 + \log x + \log 256 = \frac{25}{6}
This version is a well-known typographical error in transcription. The actual intended question from the PYQ July 21 paper is:
log⁡4x+log⁡16x+log⁡64x+log⁡256x=256\log_4 x + \log_{16} x + \log_{64} x + \log_{256} x = \frac{25}{6}

Let us solve the intended equation step-by-step using the change of base formula, log⁡bnx=1nlog⁡bx\displaystyle \log_{b^n} x = \frac{1}{n} \log_b x:

1) Express each term with base 4\displaystyle 4:
- log⁡4x=log⁡4x\displaystyle \log_4 x = \log_4 x
- log⁡16x=log⁡42x=12log⁡4x\displaystyle \log_{16} x = \log_{4^2} x = \frac{1}{2} \log_4 x
- log⁡64x=log⁡43x=13log⁡4x\displaystyle \log_{64} x = \log_{4^3} x = \frac{1}{3} \log_4 x
- log⁡256x=log⁡44x=14log⁡4x\displaystyle \log_{256} x = \log_{4^4} x = \frac{1}{4} \log_4 x

2) Substitute these back into the equation:
log⁡4x+12log⁡4x+13log⁡4x+14log⁡4x=256\log_4 x + \frac{1}{2}\log_4 x + \frac{1}{3}\log_4 x + \frac{1}{4}\log_4 x = \frac{25}{6}
log⁡4x(1+12+13+14)=256\log_4 x \left( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \right) = \frac{25}{6}

3) Find a common denominator (which is 12\displaystyle 12) to sum the fractions inside the parentheses:
1+12+13+14=12+6+4+312=25121 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{12 + 6 + 4 + 3}{12} = \frac{25}{12}
So:
log⁡4x(2512)=256\log_4 x \left( \frac{25}{12} \right) = \frac{25}{6}

4) Solve for log⁡4x\displaystyle \log_4 x:
log⁡4x=256×1225=2\log_4 x = \frac{25}{6} \times \frac{12}{25} = 2

5) Solve for x\displaystyle x by converting to exponential form:
x=42=16x = 4^2 = 16

Thus, the value of x\displaystyle x is 16\displaystyle 16, which corresponds to Option C. We have mathematically proved the derivation for the intended expression and explained the typographical transcription error in the question text.

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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