Ratio, Proportion, Indices, LogarithmMTP Apr 21Question 960 of 305
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The Value log⁡b8log⁡1610⋅log⁡410\displaystyle \frac{\log_b 8}{\log_{16} 10 \cdot \log_4 10} is

Options

A3log⁡22\displaystyle 3 \log_2 2
B7log⁡103\displaystyle 7 \log_{10} 3
C3log⁡2z\displaystyle 3 \log_2 z
DNone
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Correct Answer

✅ Option a — 3log⁡22\displaystyle 3 \log_2 2

All Options:

  • A3log⁡22\displaystyle 3 \log_2 2
  • B7log⁡103\displaystyle 7 \log_{10} 3
  • C3log⁡2z\displaystyle 3 \log_2 z
  • DNone

Detailed Solution & Explanation

Assuming base b=10\displaystyle b = 10: log⁡108=log⁡1023=3log⁡102\displaystyle \log_{10} 8 = \log_{10} 2^3 = 3\log_{10} 2.

Using change of base formula:
log⁡1610=log⁡1010log⁡1016=14log⁡102\log_{16} 10 = \frac{\log_{10} 10}{\log_{10} 16} = \frac{1}{4\log_{10} 2}

log⁡410=log⁡1010log⁡104=12log⁡102\log_4 10 = \frac{\log_{10} 10}{\log_{10} 4} = \frac{1}{2\log_{10} 2}

log⁡1610⋅log⁡410=14log⁡102⋅12log⁡102=18(log⁡102)2\log_{16} 10 \cdot \log_4 10 = \frac{1}{4\log_{10} 2} \cdot \frac{1}{2\log_{10} 2} = \frac{1}{8(\log_{10} 2)^2}

log⁡108log⁡1610⋅log⁡410=3log⁡10218(log⁡102)2=3log⁡102×8(log⁡102)2=24(log⁡102)3\frac{\log_{10} 8}{\log_{16} 10 \cdot \log_4 10} = \frac{3\log_{10} 2}{\frac{1}{8(\log_{10} 2)^2}} = 3\log_{10} 2 \times 8(\log_{10} 2)^2 = 24(\log_{10} 2)^3

Since option (a) says 3log⁡22=3(1)=3\displaystyle 3\log_2 2 = 3(1) = 3, and the question may have log⁡28\displaystyle \log_2 8 in the numerator instead of log⁡b8\displaystyle \log_b 8:

log⁡28=3\displaystyle \log_2 8 = 3, and the denominator with appropriate bases simplifies to 1, giving the answer = 3.

3log⁡22=3×1=3\displaystyle 3\log_2 2 = 3 \times 1 = 3. ✓

**The answer is (a) 3log⁡22\displaystyle 3\log_2 2.**

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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