Question

The equation $x^{[{\left({\log }_{5}x\right)}^2-(9/2){\log }_{5}x+5]}=5\sqrt{5}$ has

• A. exactly three real solutions
• B. at least one real solutions
• C. exactly one irrational solution
• D. complex roots

Answer 1

Answer: (A), (B), (C)

The correct options are
A exactly three real solutions
B at least one real solutions
C exactly one irrational solution

$x^{[{\left({\log }_{5}x\right)}^2-(9/2){\log }_{5}x+5]}=5\sqrt{5}$ is valid when $x>0$
Taking logarithms with base 5 on both the sides, we get
$[{\left({\log }_{5}x\right)}^2-\frac{9}{2}{\log }_{5}x+5]{\log }_{5}x=\frac{3}{2}[\because \log a^m=m\log a\text{\&}{\log }_{a}a=1]$
$\Rightarrow 2{\left({\log }_{5}x\right)}^3-9{\left({\log }_{5}x\right)}^2+10{\log }_{5}x-3=0$
$\Rightarrow ({\log }_{5}x-1)({\log }_{5}x-3)(2{\log }_{5}x-1)=0$
$\Rightarrow {\log }_{5}x=1,{\log }_{5}x=3,2{\log }_{5}x=1$
$\Rightarrow x=5,125,\sqrt{5}$
Ans: A,B,C