Recognize that e² implies a natural logarithm, meaning the base is e. So, e² = _e e².

$$
\log_e e^2 = 2 \log_e e
$$

$$
2 \log_e e = 2 \times 1
$$
$$
= 2
$$
The value of $\log e^2$ is $\boxed{\text{2}}$.

Alternatively, use the definition $\tan \theta = \frac{\sin \theta}{\cos \theta}$.  
We know that $\sin 45^\circ = \frac{\sqrt{2}}{2}$ and $\cos 45^\circ = \frac{\sqrt{2}}{2}$.

$$
\tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}
$$

$$
= 1
$$
The value of $\tan 45^\circ$ is $\boxed{\text{1}}$.