Ratio, Proportion, Indices, LogarithmMTP Jun 23 Series IQuestion 969 of 305
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If 12log⁡104=y\displaystyle \frac{1}{2} \log_{10} 4 = y and if 12log⁡109=x\displaystyle \frac{1}{2} \log_{10} 9 = x, then the value of log⁡1015\displaystyle \log_{10} 15

Options

Ax−y+1\displaystyle x - y + 1
Bx+y−1\displaystyle x + y - 1
Cx+y+1\displaystyle x + y + 1
Dy−x+1\displaystyle y - x + 1
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Correct Answer

✅ Option a — x−y+1\displaystyle x - y + 1

All Options:

  • Ax−y+1\displaystyle x - y + 1
  • Bx+y−1\displaystyle x + y - 1
  • Cx+y+1\displaystyle x + y + 1
  • Dy−x+1\displaystyle y - x + 1

Detailed Solution & Explanation

Given:

y=12log⁡104=12log⁡1022=log⁡102y = \frac{1}{2}\log_{10} 4 = \frac{1}{2}\log_{10} 2^2 = \log_{10} 2

x=12log⁡109=12log⁡1032=log⁡103x = \frac{1}{2}\log_{10} 9 = \frac{1}{2}\log_{10} 3^2 = \log_{10} 3

Now:

log⁡1015=log⁡10(3×5)=log⁡103+log⁡105\log_{10} 15 = \log_{10}(3 \times 5) = \log_{10} 3 + \log_{10} 5

=log⁡103+log⁡10102=log⁡103+log⁡1010−log⁡102= \log_{10} 3 + \log_{10} \frac{10}{2} = \log_{10} 3 + \log_{10} 10 - \log_{10} 2

=x+1−y=x−y+1= x + 1 - y = x - y + 1

**The answer is (a) x−y+1\displaystyle x - y + 1.**

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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