CAT 2025 Slot 3 QA Question 4: f $x$ is a positive real number such that $4\log_{10} x + 4\log_{100} x + 8\log_{1000} x = 13$, then the greatest integer not exceeding $x$, is

Question

f $x$ is a positive real number such that $4\log_{10} x + 4\log_{100} x + 8\log_{1000} x = 13$, then the greatest integer not exceeding $x$, is

Accepted Answer

Let $\log_{10}x = t$.
Then $\log_{100}x = \dfrac{t}{2}$ and $\log_{1000}x = \dfrac{t}{3}$.
$4t + 4\left(\dfrac{t}{2}\right) + 8\left(\dfrac{t}{3}\right) = 13$
$\Rightarrow 4t + 2t + \dfrac{8t}{3} = 13$
$\Rightarrow \dfrac{26t}{3} = 13$
$\Rightarrow t = \dfrac{3}{2}$
So, $x = 10^{3/2} = 10\sqrt{10} \

f xxx is a positive real number such that 4log⁡10x+4log⁡100x+8log⁡1000x=134\log_{10} x + 4\log_{100} x + 8\log_{1000} x = 134log10​x+4log100​x+8log1000​x=13, then the greatest integer not exceeding xxx, is

f $x$ is a positive real number such that $4\log_{10} x + 4\log_{100} x + 8\log_{1000} x = 13$ , then the greatest integer not exceeding $x$ , is