$\max_i x_i\ \leq \log \sum_{i} e^{x_i}$?

Question

Let $x = (x_1 ,\ldots, x_n)$ be a vector, then

$$\max_i x_i\ \leq \log \sum_i e^{x_i}.$$

Is this correct? How to see that?

Answer 1

Yes, because, for each $i$,$$x_j=\log\left(e^{x_j}\right)\leqslant\log\left(\sum_ie^{x_i}\right).$$

Answer 2

Hints: [Yes, the claim is correct]

• $\log(x)$ is an increasing function
• $e^x$ only attains positive values.
• $\log(e^{x_i})=x_i$