Question: what is domain of this function?

Question

Question: what is domain of this function?

F(x, y)= (ln(x-6)) ^ln(y-5), x and y € R

Answer 1

The argument of a log function can never be less than or equal to zero. Therefore, you must have x - 6 > 0 ==> x > 6 and y - 5 > 0 ==> y > 5.

In addition, a negative number raised to non-integral powers often returns non-real results. For example, (-4)^(1/2) is 2i, an imaginary result, and we get this value from x = e^(-4) + 6 (which is approximately 6.018) and y = sqrt(e) + 5 ( which is approximately 6.65), which falls within the domain we just defined. Therefore, you also need ln(x - 6) >= 0 ==> x - 6 >= 1 ==> x >= 7, which is a subset of the previous requirement that x > 6 and as a result is considered more strict and can simply replace it. Therefore, the basic answer is the intersection of x >= 7 and y > 5.

However, values of x such that 6 < x < 7 are acceptable as long as ln(y - 5) is an integer or is rational where its denominator is odd. For example, if x = 6.5 and y = e^(1/3) + 5 (which is approximately 6.40), we get F(x, y) = (ln(6.5 - 6))^(ln(e^(1/3) + 5 - 5)) = (ln(0.5))^(ln(e^(1/3)) = (-0.69)^(1/3) = -0.88 (rounded), a real result. So (6.5, 6.40) falls within the domain, even though it doesn't satisfy x >= 7.

Answer 2

y-5>0 so y>5
x-6>0 so x>0 but also ln(x-6)>0 so x-6>1 and x>The domain is x>7 and y >5