find the domain of root of a logarithmic function 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?

First of all, I would suggest to write your $\log$ in a fixed basis $$\log_x(10-x^2) = \frac{\ln(10-x^2)}{\ln(x)}.$$
This changes your question to $$\frac{\ln(10-x^2)}{\ln(x)} > 0$$
or equivalently: $$\ln(x)\ln(10-x^2) > 0.$$
Now, you have to find all roots of $10-x^2$ which gives you the intervals where $\ln(10-x^2) > 0$ and then you are almost finished.
