Logarithms and Linear Equations
DAILY_CHALLENGE
Grade None
Question:
Let $a=3\sqrt{2}$ and $b=\dfrac{1}{5^{1/6}\sqrt{6}}$. If $x,y\in\mathbb{R}$ are such that
$$3x+2y=\log_a(18)^{5/4} \quad \text{and} \quad 2x-y=\log_b(\sqrt{1080}),$$
then $4x+5y$ is equal to ___.
Step-by-Step Solution
Key Concept: Recognise a² = 18 and b⁻⁶ = 1080 to convert logs to simple values
$a=3\sqrt{2}$: $a^2=18$, so $\log_a 18=2$. Thus $\log_a(18^{5/4})=\dfrac{5}{4}\cdot2=\dfrac{5}{2}$.
$b=5^{-1/6}\cdot6^{-1/2}$: $b^{-6}=5\cdot6^3=1080$, so $b^{-3}=\sqrt{1080}$. Thus $\log_b(\sqrt{1080})=-3$.
System: $3x+2y=\dfrac{5}{2}$ and $2x-y=-3$.
From second: $y=2x+3$. Substitute: $3x+2(2x+3)=\dfrac{5}{2}\Rightarrow7x=\dfrac{5}{2}-6=-\dfrac{7}{2}\Rightarrow x=-\dfrac{1}{2}$.
$y=2(-\frac{1}{2})+3=2$.
$4x+5y=4(-\frac{1}{2})+5(2)=-2+10=8$.
Correct Answer: 8