Logarithmic and exponential equations dpp JEE Mains and Advance

JEE Maths DPP

Logarithmic and Exponential Equations (Free PDF)

DPP: LOG-EQ-2026-003

Part A: Multiple Choice Questions (Q1–Q10)

  1. The number of solutions of the equation $\log_4(x – 1) = \log_2(x – 3)$ is:
    1. 3
    2. 1
    3. 2
    4. 0
  2. If $3^x \cdot 2^{x+1} = 100$, then $x$ is equal to:
    1. $\log_6(50)$
    2. $\frac{2 – \log_{10} 2}{\log_{10} 6}$
    3. $\frac{\log 100}{\log 6}$
    4. $\frac{2}{\log_{10} 6} – 1$
  3. Solve for $x$: $2\log_x a + \log_{ax} a + 3\log_{a^2x} a = 0$:
    1. $a^{-4/3}$
    2. $a^{1/2}$
    3. $a^{-1/2}$
    4. $a^{-3/4}$
  4. The sum of the solutions of the equation $x^{\log_3 x} = 9$ is:
    1. $10/3$
    2. $9$
    3. $82/9$
    4. $4$
  5. If $\ln(x^2 – 1) = \ln(x-1) + \ln 2$, then the value of $x$ is:
    1. 1
    2. 2
    3. 3
    4. No solution
  6. The solution set of $5^{x+1} + 5^{1-x} = 26$ is:
    1. $\{1, -1\}$
    2. $\{0, 1\}$
    3. $\{1, 2\}$
    4. $\{0, -1\}$
  7. If $\log_{10}(x-1)^3 – 3\log_{10}(x-3) = \log_{10} 8$, then $x$ equals:
    1. 5
    2. 4
    3. 3
    4. 2
  8. Number of real solutions of $\sqrt{\log_{10}(-x)} = \log_{10}\sqrt{x^2}$ is:
    1. 1
    2. 2
    3. 3
    4. 0
  9. If $e^{2x} – 3e^x + 2 = 0$, then $x$ is:
    1. $\ln 1, \ln 2$
    2. $0, \ln 2$
    3. $1, 2$
    4. Both (a) and (b)
  10. The product of roots of the equation $\ln^2 x – 3\ln x + 2 = 0$ is:
    1. $e^3$
    2. $e^2$
    3. 3
    4. $e$

Part B: Integer Answer (Q11–Q13)

  1. The value of $x$ satisfying $\log_3(5 + 4\log_3(x-1)) = 2$ is:
    1. 2
    2. 4
    3. 10
    4. 28
  2. If $x^{\frac{3}{4}(\log_2 x)^2 + \log_2 x – \frac{5}{4}} = \sqrt{2}$, then the number of solutions is:
    1. 1
    2. 2
    3. 3
    4. 0
  3. Solve $7^{\log_x 2} = 2$:
    1. 7
    2. 2
    3. 14
    4. $e$

Part B: Subjective (Q14–Q15)

  1. Solve the simultaneous equations for $x$ and $y$: $$ \log_{10} x + \frac{1}{2}\log_{10} x + \dots = y $$ and $$ x^y = 10^8. $$
  2. Find the value of $x$ satisfying the equation $$ 4^x – 3^{x – 1/2} = 3^{x + 1/2} – 2^{2x-1}. $$

Answer Key

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10
B B A A A A D A A B
Q11 Q12 Q13 Q14 Q15
B C A (100, 4) or (0.01, -4) $\frac{3}{2}$