Warm Up Solutions

Simplify.

${(e^{2})}^{x}$

$e^{2 \cdot x} e^{2 x}$

$e^{x} \cdot e^{x}$

$e^{x + x} e^{2 x}$

Note

Q: What exponent rule is used for problem 1? For problem 2?

A: Problem 1 is the power rule, and problem 2 is the product rule.

Q: Why do problems 1 and 2 have the same answer?

A: Because using the exponent rules, $2 \cdot x$ and $x + x$ result in the same answer.

Solve.

$\log_{3} (x - 3) + \log_{3} (x + 3) = 3$

$\begin{array}{rcl} \log_{3} (x - 3) (x + 3) & = & 3 \\ \log_{3} (x^{2} - 9) & = & 3 \\ 3^{3} & = & x^{2} - 9 \\ 27 & = & x^{2} - 9 \\ 0 & = & x^{2} - 36 \\ 0 & = & (x - 6) (x + 6) \\ x & = & - 6, 6 \\ x & = & 6 \end{array}$

Note

Q: What rule do you use to write the left side of the equation with one log?

A: product rule