Ratio, Proportion, Indices, LogarithmMCQPYQ May 19Question 953 of 305
All Questions

If a=log⁡2412\displaystyle a = \log_{24} 12, b=log⁡3624\displaystyle b = \log_{36} 24, log⁡4836\displaystyle \log_{48} 36 then prove that 1+abc=\displaystyle 1 + abc =

Options

A2bc\displaystyle 2bc
B2ca\displaystyle 2ca
C2ba\displaystyle 2ba
D2ab\displaystyle 2ab
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option a — 2bc\displaystyle 2bc

All Options:

  • A2bc\displaystyle 2bc
  • B2ca\displaystyle 2ca
  • C2ba\displaystyle 2ba
  • D2ab\displaystyle 2ab

Ad

Detailed Solution & Explanation

Let a=log⁡2412\displaystyle a = \log_{24} 12, b=log⁡3624\displaystyle b = \log_{36} 24, c=log⁡4836\displaystyle c = \log_{48} 36.

Using change of base formula, let all logs be natural logs:

a=ln⁡12ln⁡24,b=ln⁡24ln⁡36,c=ln⁡36ln⁡48a = \frac{\ln 12}{\ln 24}, \quad b = \frac{\ln 24}{\ln 36}, \quad c = \frac{\ln 36}{\ln 48}

abc=ln⁡12ln⁡24⋅ln⁡24ln⁡36⋅ln⁡36ln⁡48=ln⁡12ln⁡48abc = \frac{\ln 12}{\ln 24} \cdot \frac{\ln 24}{\ln 36} \cdot \frac{\ln 36}{\ln 48} = \frac{\ln 12}{\ln 48}

1+abc=1+ln⁡12ln⁡48=ln⁡48+ln⁡12ln⁡48=ln⁡(48×12)ln⁡48=ln⁡576ln⁡481 + abc = 1 + \frac{\ln 12}{\ln 48} = \frac{\ln 48 + \ln 12}{\ln 48} = \frac{\ln(48 \times 12)}{\ln 48} = \frac{\ln 576}{\ln 48}

Now 576=242\displaystyle 576 = 24^2, so ln⁡576=2ln⁡24\displaystyle \ln 576 = 2\ln 24:

1+abc=2ln⁡24ln⁡481 + abc = \frac{2\ln 24}{\ln 48}

Now 2bc=2⋅ln⁡24ln⁡36⋅ln⁡36ln⁡48=2ln⁡24ln⁡48\displaystyle 2bc = 2 \cdot \frac{\ln 24}{\ln 36} \cdot \frac{\ln 36}{\ln 48} = \frac{2\ln 24}{\ln 48}.

Therefore 1+abc=2bc\displaystyle 1 + abc = 2bc. ✓

**The answer is (a) 2bc\displaystyle 2bc.**

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.

More Questions from Ratio, Proportion, Indices, Logarithm

3x−2:5x+6\displaystyle 3x-2:5x+6 the duplicate ratio of 2:3\displaystyle 2:3 then find the value x\displaystyle x.

If the ratio of two numbers is 7:11\displaystyle 7:11. If 7\displaystyle 7 is added to each number then the new ratio will be 2:3\displaystyle 2:3 then the numbers are.

If x:y=z:7=4:11\displaystyle x:y=z:7=4:11 then x+y+zz\displaystyle \frac{x+y+z}{z} is

The ratio of two numbers are 3:4\displaystyle 3:4. The difference of their squares is 28\displaystyle 28 greater is:

The price of scooter and moped are in the ratio 7:9\displaystyle 7:9. The price of moped is ₹1,600\displaystyle ₹1,600 more than that of scooter. Then the price of moped is:

If A:B=3:7\displaystyle A:B=3:7, then 3a+2b:4a+5b=?\displaystyle 3a+2b:4a+5b=?

Ready to Master Ratio, Proportion, Indices, Logarithm?

Practice all 305 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free