The area between the graphs $f_t(x)=e^{x-t}-t$ and $g_t(x)=\log(x+t)+t$ is irrational only if $t$ is irrational. 
Find the smallest $t \in \mathbb{R}$ such that the graphs of $f_t:x\mapsto \exp(x-t)-t$ and $g_t:x\mapsto\log(x+t)+t$ intersect. Note that $\log$ is meant as the natural logarithm. 

Hint: I guess it's $\frac{1}{2}$.
Now the real problem (it's hard): 

Prove or disprove: whenever the area $A$ between $f_t$ and $g_t$ exists and $A \in \mathbb{R} \setminus \mathbb{Q}$ then $t\in \mathbb{R} \setminus \mathbb{Q}$.


Assume it to be true, then
If $t \in \mathbb{Q} \Rightarrow $  Area between $f$ and $g$ $\in \mathbb{Q}$
Then specially for $t = 0$ 
$$
\int_{0}^{\infty} e^{x}- \log(x) dx = \left[x+  e^x + x (-\log (x)) \right]_{0}^{\infty} = \\
 \lim_{x \to \infty} x +  e^x + x (-\log (x)) - \lim_{x \ \to 0 } x+  e^x + x (-\log (x)) = \infty
$$
