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When finding a domain you look to prevent certain things from happening. The most common of these are:
Zero denominators
Arguments or bases to logarithms that are zero or negative
Negative radicands (expressions within a radical) of even-numbered roots.
Your equation has no denominators but it does have a logarithm and an even-numbered root (square roots are 2nd roots).
The base of the logarithm is e which is a positive number so we're OK on the base. The argument of the logarithm is a square root. Square roots are never negative but they can be zero! So we have to avoid a zero argument. A square root is zero only if the radicand is zero. So only if x = 0. This means 0 cannot be in the domain because it would make the argument of the logarithm zero.
We must also make sure that the radicand of your square root is not negative. In other words the radicand must be greater than or equal to zero. Since the radicand is x, x must be greater than or equal to zero.
So between the two, "x is not zero" (because of the logarithm) and "x is greater than or equal to zero" from the square root, we get a domain of:
"x is greater than zero":
This domain ensures that the radicand of the square root is never negative and the argument of the logarithm is always positive.
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