Here are the solutions to the given questions: 1. $\log e^2$ Step 1: Recognize that $\log e^2$ implies a natural logarithm, meaning the base is $e$. So, $\log e^2 = \log_e e^2$. Step 2: Apply the logarithm property $\log_b (x^y) = y \log_b x$. $$ \log_e e^2 = 2 \log_e e $$ Step 3: Apply the logarithm property $\log_b b = 1$. $$ 2 \log_e e = 2 \times 1 $$ $$ = 2 $$ The value of $\log e^2$ is $\boxed{\text{2}}$. 2. $\tan 45^\circ$ Step 1: Recall the standard trigonometric value for $\tan 45^\circ$. Alternatively, use the definition $\tan \theta = \frac{\sin \theta}{\cos \theta}$. We know that $\sin 45^\circ = \frac{\sqrt{2}}{2}$ and $\cos 45^\circ = \frac{\sqrt{2}}{2}$. Step 2: Substitute the values into the formula. $$ \tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} $$ Step 3: Simplify the expression. $$ = 1 $$ The value of $\tan 45^\circ$ is $\boxed{\text{1}}$.