Questions about an inequality

Question

Let $\alpha>0$. Is the following inequality $$\log^\alpha(1+x)\leq x$$ true for $x>0$? What about when $\alpha>1$?

Answer 1

Let $$f(x)=\ln 1(1+x)-x \implies\frac{1}{1+x}-1=-\frac{x}{1+x} < \forall~ x \ge 0$$ So $f(x)$ is a decreasing function for $x\ge 0$, then $$f(x) \le f(0) \implies f(x) \le 0 \implies \ln(1+x) \le x.$$ $$a>1 \implies \ln^{a}(1+x) < \ln(1+x) \le x.$$

Answer 2

Counterexample:

$\ln(1+\mathrm e)\approx 1.313\;$ and $\;1.313^6>1+6\cdot 0.313=2.878>\mathrm e$.