JEE Math DPP Fundamentals of Logarithm

JEE Maths DPP

Fundamentals of Logarithms

DPP : LOG-F-2026-SET01

Part A: Multiple Choice Questions (Single Correct)

  1. The value of $\log_{\sqrt{3}} (27)$ is equal to:
    1. 3
    2. 6
    3. 9
    4. 1.5
  2. If $a = \log_{24} 12$, $b = \log_{36} 24$, and $c = \log_{48} 36$, then the value of $1 + abc$ is:
    1. $2bc$
    2. $2ac$
    3. $2ab$
    4. $ab$
  3. Let $x = (0.15)^{20}$. Then the number of zeros between the decimal point and the first non-zero digit in $x$ is (Given $\log_{10} 2 = 0.301, \log_{10} 3 = 0.477$):
    1. 15
    2. 16
    3. 17
    4. 18
  4. The value of $81^{1/\log_5 3} + 27^{\log_9 36} + 3^{4/\log_7 9}$ is:
    1. 890
    2. 930
    3. 750
    4. 810
  5. If $\log_{10} 2 = a$ and $\log_{10} 3 = b$, then $\log_5 12$ in terms of $a$ and $b$ is:
    1. $\frac{2a+b}{1-a}$
    2. $\frac{a+2b}{1-a}$
    3. $\frac{2a+b}{1+a}$
    4. $\frac{a+b}{1-a}$
  6. The solution set of the equation $\log_4 (x^2 – 1) = \log_4 (x-1)$ is:
    1. $\{0, 1\}$
    2. $\{-1\}$
    3. $\{1\}$
    4. $\emptyset$
  7. If $x = \log_a (bc)$, $y = \log_b (ca)$, and $z = \log_c (ab)$, then the value of $\frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1}$ is:
    1. 0
    2. 1
    3. 2
    4. $abc$
  8. 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. 1
    4. 4
  9. If $\ln(x)$ represents the natural logarithm, what is the value of $e^{2\ln 3 + \ln 5}$?
    1. 15
    2. 30
    3. 45
    4. 25
  10. The number of real solutions of $\log_2 x + \log_x 2 = 2$ is:
    1. 0
    2. 1
    3. 2
    4. Infinite
  11. The domain of $f(x) = \log_{10}(1-x)$ is:
    1. $(-\infty, 1)$
    2. $(1, \infty)$
    3. $(-\infty, 0)$
    4. $\mathbb{R}$
  12. If $7^{\log_7 (x^2 – 4x + 5)} = x – 1$, then $x$ is:
    1. 2
    2. 3
    3. 2, 3
    4. None of these
  13. The characteristic of $\log_{10} (0.000345)$ is:
    1. -3
    2. -4
    3. 4
    4. 3

Part B: Subjective Questions

  1. Solve for $x$: $\log_2 (\log_3 (\log_4 x)) = 0$.
  2. Prove that $a^{\log b} = b^{\log a}$ where the base of logarithm is any positive real number $c \neq 1$.

Part C: Integer Answer Type

  1. Find the value of $x$ satisfying $$ \log_3 x + \log_9 x + \log_{27} x = 11. $$
  2. If $\log_{10} 2 = 0.3010$, find the number of digits in $2^{100}$.
  3. Find the value of $$ 5^{\log_5 7} + \log_2 128 – \ln e^3. $$
  4. If $\log_{2} 8 = a$, then find the value of $\log_{4} 64$ in terms of $a$ and calculate it when $a=3$. (Final integer answer required for $a=3$).
  5. Find the sum of all values of $x$ satisfying $$ \log_{10} x^2 = (\log_{10} x)^2. $$

Part D: Assertion Reason Type

  1. Assertion (A): $\log_2 3$ is an irrational number.
    Reason (R): If $\log_a b = \frac{p}{q}$, then $a^p = b^q$.
    1. Both A and R are true and R is the correct explanation of A.
    2. Both A and R are true but R is NOT the correct explanation of A.
    3. A is true but R is false.
    4. A is false but R is true.
  2. Assertion (A): The value of $\log_{0.5} 4 = -2$.
    Reason (R): $\log_a x$ is defined only when $a > 0, a \neq 1$ and $x > 0$.
    1. Both A and R are true and R is the correct explanation of A.
    2. Both A and R are true but R is NOT the correct explanation of A.
    3. A is true but R is false.
    4. A is false but R is true.

Answer Key

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11
(b) (a) (b) (a) (d) (b) (b) (b) (c) (b) (a)
Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22
(a) (b) 64 Proof 729 31 11 $3$ 101 (b) (b)