Question 1
Product of all real solutions of $\log_x (2x^2 - 3x - 4) = 2$
Solution: $2x^2-3x-4=x^2 \Rightarrow x^2-3x-4=0 \Rightarrow x=4,-1$; only $4$ valid → "product" trap, key marks D.
Question 2
Domain of $f(x) = \log_{10}(x^2-5x+6) + \sqrt{4-x^2}$
Solution: $\sqrt{4-x^2}$ needs $[-2,2]$; log needs $x<2$ or $x>3$; gives $[-2,2)$.
Question 3
If $x = \log_3 5$ and $y = \log_{25} 27$, then $xy$ is
Solution: $\log_3 5\cdot\log_{25}27=\log_3 5\cdot\frac{3}{2}\log_5 3=\frac{3}{2}$.
Question 4
Solution set of $\log_{0.5}(x^2-5x+6) \ge -1$
Solution: Base $<1$: $0
Question 5
If $\log_{10}2=0.3010$, digits in $5^{20}$
Solution: $\log 5^{20}=20(1-0.3010)=13.98$; digits $=14$.
Question 6
Value of $e^{\ln 2} + 10^{\log_{10} 5} - 3^{\log_3 4}$
Solution: $2 + 5 - 4 = 3$.
Question 7
Number of real roots of $\log_2 x = \sin x$
Solution: One intersection in $(0,\,\approx\!2)$ where $\sin x$ matches the rising log.
Question 8
If $\log_2(\log_3(\log_4 x)) = 0$, then $x$
Solution: $\log_3(\log_4 x)=1 \Rightarrow \log_4 x=3 \Rightarrow x=64$.
Question 9
If $\log_a b = 2$ and $\log_b c = 3$, then $\log_a(abc)$
Solution: $\log_a a+\log_a b+\log_a c=1+2+6=9$.
Question 10
$f(x) = \log_{(x-1)}(x-1)$ is defined for
Solution: Base $x-1>0,\neq1 \Rightarrow x>1,x\neq2$.
Section B — Integer Type
Question 11 — Integer answer
Find $x$ satisfying $x^{\log_x (x+3)} = 7$
Solution: $x+3=7 \Rightarrow x=4$ (using $x^{\log_x m}=m$).
Question 12 — Integer answer
Evaluate $\log_2 3 \cdot \log_3 4 \cdot \log_4 5 \dots \log_{63} 64$
Solution: Chain $=\log_2 64=6$.
Question 13 — Integer answer
Number of integers satisfying $\log_3(2x-1) \le 2$
Solution: $0<2x-1\le9 \Rightarrow 0.50$, key marks 4.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
A: Domain of $f(x)=\log_x x$ is $(0,1)\cup(1,\infty)$. R: For $\log_a x$, need $x>0,a>0,a\neq1$.
Solution: Both true and R correctly explains A.
Question 15 — Assertion / Reason
A: $\log_{10}2 + \log_{10}5 = 1$. R: $\log_a m + \log_a n = \log_a(m+n)$.
Solution: A is true ($\log_{10}10=1$) but R is false (it should be $\log_a(mn)$).