Ratio, Proportion, Indices, LogarithmMCQMTP Jun 23 Series IQuestion 968 of 305
All Questions

If log⁡34⋅log⁡45⋅log⁡56⋅log⁡67⋅log⁡78⋅log⁡89=x\displaystyle \log_3 4 \cdot \log_4 5 \cdot \log_5 6 \cdot \log_6 7 \cdot \log_7 8 \cdot \log_8 9 = x, then find the value of x\displaystyle x

Options

A4\displaystyle 4
B2\displaystyle 2
C3\displaystyle 3
D1\displaystyle 1
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option b — 2\displaystyle 2

All Options:

  • A4\displaystyle 4
  • B2\displaystyle 2
  • C3\displaystyle 3
  • D1\displaystyle 1

Ad

Detailed Solution & Explanation

Using the chain rule of logarithms (change of base formula), each consecutive pair cancels:

log⁡34⋅log⁡45⋅log⁡56⋅log⁡67⋅log⁡78⋅log⁡89\log_3 4 \cdot \log_4 5 \cdot \log_5 6 \cdot \log_6 7 \cdot \log_7 8 \cdot \log_8 9

=ln⁡4ln⁡3⋅ln⁡5ln⁡4⋅ln⁡6ln⁡5⋅ln⁡7ln⁡6⋅ln⁡8ln⁡7⋅ln⁡9ln⁡8= \frac{\ln 4}{\ln 3} \cdot \frac{\ln 5}{\ln 4} \cdot \frac{\ln 6}{\ln 5} \cdot \frac{\ln 7}{\ln 6} \cdot \frac{\ln 8}{\ln 7} \cdot \frac{\ln 9}{\ln 8}

All intermediate terms cancel (telescoping product):

=ln⁡9ln⁡3=log⁡39=log⁡332=2= \frac{\ln 9}{\ln 3} = \log_3 9 = \log_3 3^2 = 2

**The answer is (b) x=2\displaystyle x = 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.

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