Question

Find the largest subset of the real numbers such that the function
log[1 − log(x^2 − 5x + 16)] is defined.

Answer 1

The largest subset of the real numbers is $(2, 3)$.

The domain of a logarithm of the form $y = \log x$ is $(0, +\infty)$. Hence, we have the following two conditions:

$1 - \log (x^{2}-5\cdot x + 16) > 0$ (1)

$x^{2}-5\cdot x + 16 > 0$ (2)

By (1):

$\log (x^{2}-5\cdot x +16) < 1$

$x^{2}-5\cdot x +16 < 10$

$x^{2}-5\cdot x +6 < 0$

$(x-3)\cdot (x-2) < 0$

The subset of (1) is $(2, 3)$.

By (2) we see that roots are conjugated complex numbers. Thus, we find that domain is represented by $\mathbb{R}^{+}$ and the largest subset of the real numbers is $(2, 3)$.

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