Laws and Properties of Logarithm Free JEE Math Dpp

JEE Maths DPP

Laws and Properties of Logarithms

DPP: LOG-PROP-2026-001

Part A: Multiple Choice Questions (Single Correct)

  1. If $\log_{10} 2 = a$ and $\log_{10} 3 = b$, then the value of $\log_{5} 12$ is:
    1. $\frac{2a+b}{1-a}$
    2. $\frac{a+2b}{1-a}$
    3. $\frac{2a+b}{1+a}$
    4. $\frac{a+b}{1-a}$
  2. The value of $3^{\log_4 5} – 5^{\log_4 3}$ is:
    1. $1$
    2. $0$
    3. $2$
    4. $\log_4 15$
  3. If $\log_2(\log_3(\log_4 x)) = 0$ and $\log_3(\log_4(\log_2 y)) = 0$, then $x + y$ is equal to:
    1. $80$
    2. $144$
    3. $80$
    4. $130$
  4. The value of $\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + \dots + \frac{1}{\log_{20} n}$ is:
    1. $\log_n (20!)$
    2. $\log_{20!} n$
    3. $\frac{1}{\log_{20!} n}$
    4. $\log_n (19!)$
  5. If $a, b, c$ are distinct positive real numbers such that $a^2 + c^2 = b^2$, then the value of $$ \frac{1}{\log_{b+c} a} + \frac{1}{\log_{b-c} a} $$ is:
    1. $1$
    2. $2$
    3. $0$
    4. $-1$
  6. The solution of the equation $\log_7 \log_5 (\sqrt{x+5} + \sqrt{x}) = 0$ is:
    1. $2$
    2. $4$
    3. $0$
    4. $1$
  7. If $\log_k A \cdot \log_5 k = 3$, then $A$ is equal to:
    1. $15$
    2. $125$
    3. $243$
    4. $k^3$
  8. Let $x = \log_{2} 3$, $y = \log_{3} 5$, $z = \log_{5} 2$. The product $xyz$ is:
    1. $0$
    2. $1$
    3. $\log_2 5$
    4. $\log_5 3$
  9. The value of $\log_3 2 \cdot \log_4 3 \cdot \log_5 4 \dots \log_{16} 15$ is:
    1. $1/2$
    2. $1/4$
    3. $4$
    4. $2$
  10. If $x = 1 + \log_a (bc)$, $y = 1 + \log_b (ca)$, and $z = 1 + \log_c (ab)$, then $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is:
    1. $0$
    2. $1$
    3. $abc$
    4. $-1$
  11. The value of $\log_2 10 – \log_8 125$ is:
    1. $\log_2 5$
    2. $1$
    3. $0$
    4. $2$
  12. If $\log_2 x + \log_4 x + \log_{16} x = \frac{21}{4}$, then $x$ equals:
    1. $8$
    2. $4$
    3. $16$
    4. $2$
  13. If $a = \log_{12} 18$ and $b = \log_{24} 54$, then $ab + 5(a-b)$ is:
    1. $1$
    2. $0$
    3. $2$
    4. $ab$

Part B: Assertion-Reason Type

Options:

  • (A) Assertion is true, Reason is true; Reason is a correct explanation for Assertion.
  • (B) Assertion is true, Reason is true; Reason is NOT a correct explanation for Assertion.
  • (C) Assertion is true, Reason is false.
  • (D) Assertion is false, Reason is true.
  1. Assertion: The value of $\log_2 3$ is an irrational number.
    Reason: If $\log_a b = p/q$ where $p, q$ are integers, then $a^p = b^q$.
  2. Assertion: $\log_2 5 > \log_3 5$.
    Reason: If $x > 1$ and $a > b > 1$, then $\log_b x > \log_a x$.

Part C: Integer Answer Type

  1. Find the value of $x$ satisfying $$ x^{\log_{10} x} = 100x. $$
  2. If $7^{\log_7 (x^2 – 4x + 5)} = x – 1$, find the sum of all real values of $x$.
  3. Find the value of $$ \log_2 128 – \log_3 243 + \log_5 125. $$
  4. If $\log_{10} 2 = 0.301$, find the number of digits in $2^{100}$.
  5. Find the value of $n$ if $$ \sum_{r=1}^n \log_2 \left( \frac{r+1}{r} \right) = 5. $$

Part D: Subjective (Advanced Style)

  1. If $x, y, z$ are in G.P., prove that $\log_a x, \log_a y, \log_a z$ are in A.P.
  2. Solve for $x$: $$ \log_2 (9 – 2^x) = 3 – x. $$

Answer Key

Q1 A Q2 B Q3 B Q4 A
Q5 B Q6 B Q7 B Q8 B
Q9 B Q10 B Q11 B Q12 A
Q13 A Q14 A Q15 A Q16 100
Q17 5 Q18 5 Q19 31 Q20 31