JEE Advanced Level

Mock Test 2 — Logarithms

15 questions • 45 minutes • auto-graded with full solutions

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Note on Test 2 Q5: $40 \times 0.4771 = 19.084$, giving characteristic $19$ and therefore $20$ digits by the "characteristic + 1" rule. The source answer key marks (A) $= 19$; the standard digit count is $20$.

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Section A — MCQ (Single Correct)

Question 1

If $\log_2 x + \log_4 x + \log_{16} x = \frac{7}{4}$, then $x$

Solution: $\log_2 x(1+\tfrac12+\tfrac14)=\tfrac74 \Rightarrow \log_2 x=1 \Rightarrow x=2$.

Question 2

Solution set of $\log_{1/3}(x^2 + x) > -1$

Solution: Base $<1$: $0

Question 3

Value of $x$ satisfying $x^{\log_3 x} = 9$

Solution: $(\log_3 x)^2=2$… gives $x=3$ or $\tfrac13$.

Question 4

Domain of $f(x) = \sqrt{\log_{0.3}\left(\frac{x-1}{x+5}\right)}$

Solution: Argument $>0$ gives $x<-5$ or $x>1$; the root condition holds throughout.

Question 5

If $\log_{10}3=0.4771$, digits in $3^{40}$

Solution: $40(0.4771)=19.084$; digits $=20$? — key marks A (19): see note.

Question 6

If $a^2 + 4b^2 = 12ab$, then $\log_e(a+2b)$ equals

Solution: $(a+2b)^2=16ab \Rightarrow \ln(a+2b)=\tfrac12(\ln a+\ln b+\ln16)$.

Question 7

Number of real solutions of $\log_2(x^2-3x+2) = \log_2(x-1)+\log_2(x-2)$

Solution: RHS needs $x>2$; LHS factors to same, but domain check leaves none valid → key A.

Question 8

$\log_2 15 \cdot \log_3 15 - \log_2 5 \cdot \log_3 5$ simplifies to

Solution: Algebraic expansion matches $\log_2 15+\log_3 15$.

Question 9

Range of $f(x) = \log_2(2 - \cos x)$

Solution: $2-\cos x\in[1,3]$, so log $\in[0,\log_2 3]$.

Question 10

If $\log_2 x + \log_x 2 = 2$, then $\log_2(x^3)$

Solution: $t+\tfrac1t=2 \Rightarrow t=1 \Rightarrow x=2$; $\log_2 8=3$.

Section B — Integer Type

Question 11 — Integer answer

Find $x$ satisfying $4^{\log_2 x} - 2x - 8 = 0$

Enter an integer value.

Solution: $4^{\log_2 x}=x^2$; $x^2-2x-8=0 \Rightarrow x=4$ (valid).

Question 12 — Integer answer

Number of integers in the domain of $f(x) = \log_{10}(9-x^2)$

Enter an integer value.

Solution: $9-x^2>0 \Rightarrow -3

Question 13 — Integer answer

Evaluate $E = \log_3(\log_2 256 - 5)$

Enter an integer value.

Solution: $\log_2 256=8$; $8-5=3$; $\log_3 3=1$.

Section C — Assertion & Reasoning

Question 14 — Assertion / Reason

A: $\log_{0.5} x > \log_{0.5} y \Rightarrow x < y$. R: $y=\log_a x$ is strictly decreasing for $a\in(0,1)$.

Solution: Both true and R correctly explains A.

Question 15 — Assertion / Reason

A: $\log_x(x-1)=2$ has exactly two real solutions. R: Converting $\log_a x=n$ to exponential form can introduce extraneous roots.

Solution: A is false (no valid solution exists) but R is true.