Question

The equation ${\left({\log }_{10}x+2\right)}^3+{\left({\log }_{10}x-1\right)}^3={\left(2{\log }_{10}x+1\right)}^3$ has

• A. no natural solution
• B. two rational solutions
• C. no prime solution
• D. one irrational solution

Answer 1

Answer: (B), (C), (D)

The correct options are
B two rational solutions
C no prime solution
D one irrational solution

Let ${\log }_{10}x+2=a$ and ${\log }_{10}x-1=b$
$\therefore a+b=2{\log }_{10}x+1$ (from the question)
Thus, the given equation(in the question) reduces to $a^3+b^3={(a+b)}^3$
$\Rightarrow 3ab(a+b)=0$
$\Rightarrow a=0$ or $b=0$ or $a+b=0$
$\Rightarrow {\log }_{10}x+2=0$ or ${\log }_{10}x-1=0$ or $2{\log }_{10}x+1=0$
$\Rightarrow x={10}^{-2}$ or $x=10$ or $x={10}^{-\frac{1}{2}}$
Hence $x=\{\frac{1}{100},10,\frac{1}{\sqrt{10}}\}$