Only One Intersection

Since the graphs intersect – actually touch – only once, the function values at the point of intersection have to match, but also the derivatives since otherwise the exponential graph would cross the line and there would be another intersection point.

We end up with the equations

$x = {\color{#D61F06}t^x} \Leftrightarrow t = {\color{#3D99F6}x^{\frac 1x}}$

$1 = t^x \ln t$

Plugging the first into the second gives

$\begin{aligned} 1 & = {\color{#D61F06}x} \cdot \ln \left( {\color{#3D99F6}x^{\frac 1x}} \right) \\ &= x \cdot \frac 1x \ln x \\ & = \ln x \\ &\Leftrightarrow x = e \end{aligned}$

Using the first equation, we get $t$ to be

$t = x^{\frac 1x} = e^{\frac 1e} \approx \boxed{1.445}$ .