[Math] find the domain of root of a logarithmic function

Question

I'm a little confused about this question since output of a logarithmic function varies from $ -\infty $ to $\infty$ .I should find the domain of this function: $ y=\sqrt{\log_x(10-x^2)} $ . How can I find the interval that makes $\log_x(10-x^2)$ greater than zero?

Answer 1

Recall for $a>0, a\neq 1,b>0$:

$$\log_ab=\frac{\ln b}{\ln a}$$

Thus we have

$$f(x)=\log_x(10-x^2)=\frac{\ln(10-x^2)}{\ln x}$$

Then $f(x)\geq 0$ if and only if $10-x^2\geq 1,x>1$ or $0<10-x^2\leq 1,0<x<1$. Since the later case cannot happen, then we must have $10-x^2\geq 1$ and $x>1$, which gives $1<x\leq3$