Is log x + log y monotonically increasing with respect to x + y?

Question

We know $\log x + \log y = \log(xy)$ which is a monotonically increasing function with respect to the product $xy$.

I am wondering if $\log x + \log y$ is still a monotonically increasing function of the sum $x+y$?

Answer 1

No.

For example, $\log 2+\log9=\log 18<\log 25=\log5+\log5$,

though $2+9=11>10=5+5$.

Answer 2

Just a little to add to the intuition. The other answer is correct.

One thing I'm confused about the question is that the query is not well-defined: log(xy) can take different values even if you know the value of x+y.

You can plot log(xy) on WolframAlpha and imagine "scanning across the graph" using hyperbolas xy=c (y=c/x) where you adjust (increase, for example) c. You can see such hyperbolas are "level sets" of f.

However, trying to do the same with diagonal lines x+y=b doesn't make sense at all.

All this fundamentally comes down to the fact that log turns multiplication into addition

Answer 3

Let $M$ be fixed and let $0 < h \le \frac M2$. Then $\lim_{h\to 0}(\ln h + \ln(M-h) = -\infty$ and $\lim_{h\to \frac M2} (\ln h + \ln(M-h)) = 2\ln \frac M2 = 2\ln M - \ln 2$.

$\ln a + \ln b$ with restriction $a + b = M$ has no lower bounded value.