If two quantities are equal, are their logarithms also equal? Consider the questions below.

1. Is it true that $4^2$ is equal to $2^4$? Is this a special case, or is $a^b$ equal to $b^a$ for any values of $a$ and $b$?

   Hint (a):

   Try a few other combinations of $a^b$ and $b^a$.

2. Is log $4^2$ equal to log $2^4$? How can you be sure?

   Hint (b):

   Simplify the exponential expressions first.

3. Are the equations $x=5$ and $\log x=\log5$ equivalent? Justify your answer.

   Hint (c):

   What solutions satisfy each equation?

4. Is the equation $\log7=\log x^2$ equivalent to the equation $7=x^2$? How can you be sure?

   Hint (d):

   See part (c).