For which value of the parameter a does the curve y=a^x

For which value of the parameter $a$ does the curve $y=a^{x}$ touch the line $y=x$? Find the point of tangency.

Step 1: The two tangency conditions

\[ a^{x_{0}}=x_{0} \quad \text{(shared point)}, \qquad a^{x_{0}}\ln a = 1 \quad \text{(shared slope).} \tag{1,2} \]

Step 2: Solve the system

Substituting (1) into (2): $x_0\ln a=1 \Rightarrow \ln a=\dfrac1{x_0} \Rightarrow a=e^{1/x_0}$. Substituting back into (1): $e^{x_0/x_0}=x_0 \Rightarrow e=x_0$.

Step 3: Verification

\[\bigl(e^{1/e}\bigr)^{e} = e^{e/e} = e = x_0\ \checkmark; \qquad a^{x_0}\ln a = e\cdot\tfrac1e = 1\ \checkmark\]