# Interpreting $e^x+e^y<e^z$

Suppose that $$(x,y,z)\in[0,1]^3$$ for simplicity. It is easy for me to interpret "$$x+y": this just means "$$z$$ is larger than the sum of x and y." Obviously "$$e^{x+y}" has the same interpretation. I'm having trouble interpreting "$$e^x+e^y" Any suggestions for an intuitive interpretation of this inequality would be greatly appreciated!

An observation: Plotting $$\left\{(x,y,z)\in[0,1]^3:e^x+e^y in MATLAB seems to suggest that in order to be in this set, $$z$$ must be larger than $$x+y$$ plus some additional buffer. Intuition behind why this happens?

• that observation is not actually true. "z must be larger than x+y" is the correct observation for $e^{x+y} < e^z$ and therefore the "buffer" is precisely the difference between $e^{x+y}$ and $e^x+e^y$. Setting $x=y=c$ and checking the extremes $c=0,1$, we note that $e^2 >2e^1$ but $e^0 < 2e^0$, so the "buffer" flips sign somewhere. Oct 31 '20 at 2:26
• It seems to be true for $x,y,z$ in $[0,1]$ but fails for larger $z$. Consider just the equation $2e^x = e^{2x}$ which has solution $e^x=2$, $z=\ln 4$ slightly more than $1$. Oct 31 '20 at 2:30
• $e^{x}+e^y <e^z$ is certainly not true when $\max(x,y) \ge z$ but it is true when $\max(x,y) +\log_e(2) \lt z$ Oct 31 '20 at 2:40
• I'm not understanding your issue. $e^x$, $e^y$ and $e^z$ are a values and if $e^x + e^y < e^z$ then ... it is so. What's to explain? If $e^x = M \in [1, e]$ and $e^y=N \in [1,e]$ and $e^z= K \in [1,e]$ and $M +N < K$ ... what needs explaining? Oct 31 '20 at 2:53
• "that observation is not actually true." Well, neither is $x + y < z$ (what if $x=7.9; y = 8.6$ and $z =0.04$?) nor is $e^{x+y} < e^z$ (ditto). Oct 31 '20 at 2:55

the whole thing with equality is a surface. Here is a picture of level curves of $$e^x + e^y$$
Note that the level curves are identical, take the one that goes through the origin and shift by vector $$(t, t).$$ In the picture use $$(\frac{1}{2}, \frac{1}{2}).$$
Here it is with some diagonal lines drawn. If you tilt your head $$45^\circ$$ so that the diagonal lines appear to be horizontal, the idea that the curves are identical becomes more believable. I do not know how to rotate a jpeg or png by $$45^\circ$$