Logarithm Change of Base Properties Free Math DPP

JEE Maths DPP

Change of Base and Evaluation

DPP: LOG-CHBASE-2026-001

Part A: Multiple Choice Questions (Q1–Q13)

  1. The value of $\log_{3} 5 \cdot \log_{25} 27$ is:
    1. $3/2$
    2. $2/3$
    3. $1$
    4. $9/4$
  2. If $a = \log_{24} 48$ and $b = \log_{36} 72$, then the value of $2ab – (a+b)$ is:
    1. $1$
    2. $0$
    3. $\log_{24} 36$
    4. None of these
  3. If $\log_{a} b = 10$, then the value of $\log_{b^5} a^2$ is:
    1. $4$
    2. $25$
    3. $1/25$
    4. $0.04$
  4. The expression $\frac{1}{\log_{3} 2} + \frac{1}{\log_{4} 2} + \dots + \frac{1}{\log_{10} 2}$ simplifies to:
    1. $\log_{2} (10!)$
    2. $\log_{2} (10!/2)$
    3. $\log_{2} (5!)$
    4. $10!$
  5. If $x = \log_{c} (ab)$, $y = \log_{a} (bc)$, and $z = \log_{b} (ca)$, then $$ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} $$ is:
    1. $1$
    2. $2$
    3. $0$
    4. $3$
  6. The value of $\log_{4} 18$ is:
    1. $1 + 2\log_{4} 3$
    2. $\frac{1}{2} + \log_{2} 3$
    3. $\frac{1}{2} + \log_{4} 9$
    4. Both (b) and (c)
  7. If $\log_{12} 27 = a$, then $\log_{6} 16$ in terms of $a$ is:
    1. $\frac{4(3-a)}{3+a}$
    2. $\frac{4(3+a)}{3-a}$
    3. $\frac{3-a}{3+a}$
    4. $\frac{a}{3-a}$
  8. Find the value of $\log_{5} \sqrt{5 \sqrt{5 \sqrt{5 \dots \infty}}}$:
    1. $1/2$
    2. $1$
    3. $0$
    4. $\infty$
  9. If $\log_{3} 2, \log_{3} (2^x – 5), \log_{3} (2^x – 7/2)$ are in A.P., then $x$ is:
    1. $2$
    2. $3$
    3. $4$
    4. $5$
  10. The value of $81^{1/\log_{5} 3} + 27^{\log_{9} 36} + 3^{4/\log_{7} 9}$ is:
    1. $890$
    2. $930$
    3. $720$
    4. $1024$
  11. If $\log_{10} 2 = 0.3010$, then the number of digits in $5^{20}$ is:
    1. $14$
    2. $15$
    3. $13$
    4. $16$
  12. Let $S = \sum_{n=2}^{100} \frac{1}{\log_{n} 100!}$. The value of $S$ is:
    1. $0$
    2. $1$
    3. $100$
    4. $1/2$
  13. If $x = \log_{2} 3 \cdot \log_{3} 4 \cdot \log_{4} 5 \dots \log_{n} (n+1) = 10$, then $n$ is:
    1. $1023$
    2. $1024$
    3. $511$
    4. $2047$

Part B: Subjective (Q14–Q15)

  1. Solve the equation: $$ 4^{\log_{9} x} – 6 \cdot 2^{\log_{9} x} + 8 = 0. $$
  2. Prove that $$ \frac{\log_{a} n}{\log_{ab} n} = 1 + \log_{a} b $$ and hence find the value of $\log_{3} 10$ if $\log_{30} 10 = a$.

Part C: Integer/Assertion Reason

  1. Assertion: $\log_{2} 3$ is irrational.
    Reason: If $\log_{a} b$ is rational, say $p/q$, then $a^p = b^q$. For $a=2, b=3$, $2^p = 3^q$ has no integer solution for $p, q > 0$.
    1. A
    2. B
    3. C
    4. D
  2. Integer: Find the value of $x$ satisfying $$ \log_{x} 2 \cdot \log_{x/16} 2 = \log_{x/64} 2. $$
  3. Integer: If $\log_{2} (\log_{2} (\log_{3} x)) = \log_{2} (\log_{3} (\log_{2} y)) = 0$, find $x-y$.
  4. Integer: Find the value of $$ 7^{\log_{7} 11} + \log_{2} 8. $$
  5. Integer: If $a^2 + b^2 = 7ab$, find the value of $k$ if $$ 2\log\left(\frac{a+b}{3}\right) = k(\log a + \log b). $$
  6. Assertion: $\log_{10} 2$ lies between $1/4$ and $1/3$.
    Reason: $2^3 < 10 < 2^4$.
  7. Integer: Calculate $$ \log_{3} 4 \cdot \log_{4} 5 \cdot \log_{5} 6 \cdot \log_{6} 7 \cdot \log_{7} 8 \cdot \log_{8} 9. $$

Answer Key

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11
A B D B A D A B B B A
Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22
B A 81, 16 $\frac{a}{1-a}$ A 4 1 14 1 A 2