In Boyd's Convex Optimization book, Example 3.25 finds the conjugate function $f^*(y):=\sup_{x\in\text{dom}(f)}(y^Tx-f(x))$ of the log-sum-exp function $f(x):=\log(\sum_{i=1}^ne^{x_i})$. First, the gradient of $y^Tx-f(x)$ is taken to yield the condition:
$$ y_i=\frac{e^{x_i}}{\sum_{j=1}^ne^{x_j}}\quad i=1,...,n $$
where we see that a solution for $y$ exists if and only if $y\succ 0$ and $\textbf{1}^Ty=1$. Then the book simply says:
By substituting the expression for $y_i$ into $y^Tx-f(x)$ we obtain $f^*(y)=\sum_{i=1}^ny_i\log(y_i)$.
So far I've been unsuccessful in deriving this. How does one proceed? All I see is:
$$ y^Tx-f(x)=\sum_{i=1}^ny_ix_i-\log(\sum_{i=1}^ne^{x_i})=\frac{\sum_{i=1}^nx_ie^{x_i}}{\sum_{j=1}^ne^{x_j}}-\log(\sum_{i=1}^ne^{x_i}) $$
But from here on I do not knonw how to proceed.