Polynomials in logarithms

Logarithmic functions are only defined for positive real values. Therefore, for $\log(P(x))$ to be defined for all real values of $x$ , we must find a polynomial $P(x)$ of degree $2$ such that $P(x) >0$ for all real values of $x$ .

Remember that a polynomial of even degree can have no real roots.

Therefore, this implies that a polynomial $P(x)$ of degree $2$ exists such that $P(x) > 0$ or $P(x)<0$ for all real values of $x$ .

Generally, all quadratic polynomials of the form

$P(x)=a(x+p)^2 + q$

where $a>0,\;q>0$ has no real roots and is positive for any real value of $x$

Proof:

We know that $(x+p)^2 \geq 0$ for all real values of $x$ .

$a > 0 \implies a(x+p)^2 \geq 0\\ q > 0 \implies a(x+p)^2+q > 0$

Examples of such polynomials:

$P(x) = x^2+1\\ P(x) = 2(x-1)^2 + 3\\ P(x) = 0.1(x+3)^2 + 0.5$

Therefore, we conclude that a polynomial $P(x)$ of degree $2$ does exist such that $\log(P(x))$ is defined for all real values of $x$ . The answer is $\boxed{\text{True}}$