Is it true always?

Given $a$ , $b$ and $\sqrt {a^2+b^2}$ are positive integers and $x$ is a real number greater than $1.3$ , which of the following statements is true?

• A : $a^x\ln {a^2}+b^x\ln {b^2}>(\sqrt {a^2+b^2})^x\ln {(a^2+b^2)}$ for all $x$
• B : $a^x\ln {a^2}+b^x\ln {b^2}>(\sqrt {a^2+b^2})^x\ln {(a^2+b^2)}$ for some $x$
• C : $a^x\ln {a^2}+b^x\ln {b^2}>(\sqrt {a^2+b^2})^x\ln {(a^2+b^2)}$ for no $x$

Notation: $\ln (\cdot) = \log_e (\cdot)$ denotes the natural logarithm .