Question

Indicate all correct alternative, where base of the log is 2. The equation $x^{\left(3/4\right)}(logx{)}^2+logx-(5/4)=\sqrt{2}$ has

• A. At least one real solution
• B. Exactly three real solution
• C. Exactly one irrational solution
• D. Complex roots

Answer 1

The correct option is A At least one real solution

$x^{\frac{3}{4}}{\left(logx\right)}^2+logx-\frac{5}{4}=\sqrt{2}$

Let $logx=ax=2a$

${(2}^{\frac{3a}{4}})a^2+a-\frac{5}{4}-\sqrt{2}=0\Delta =1+4(\frac{5}{4}+\sqrt{2}\left)\right(2^{\frac{3a}{4}})=1+(5+4\sqrt{2})2^{\left(\frac{3a}{4}\right)}>0$

Therefore, given equation has atleast one real solution.