2. It is known that the graphs of the functions $f(x)=x^{p}$ and $g(x)=\ln x$ touch (have a common point where the tangents to both graphs coincide). Find the constant $p$ and the point of tangency. (16 points)
Solution. Since the graphs have a common tangent, the following relationships hold for the abscissa of the point of tangency:

$$
\left\{\begin{array} { c } 
{ f ( x ) = g ( x ) } \\
{ f ^ { \prime } ( x ) = g ^ { \prime } ( x ) }
\end{array} \Rightarrow \left\{\begin{array} { c } 
{ x ^ { p } = \operatorname { l n } x } \\
{ p x ^ { p - 1 } = 1 / x }
\end{array} \Rightarrow \left\{\begin{array}{l}
x^{p}=\ln x \\
x^{p}=1 / p
\end{array} \Rightarrow \ln x=\frac{1}{p} \Rightarrow x=e^{1 / p}, x^{p}=e=\ln x\right.\right.\right.
$$

Thus, $x=e^{e}, p=\frac{1}{e}, y=e$.

Answer: $p=\frac{1}{e}$, point of tangency ( $e^{e} ; e$ ).