6.2 Rewriting logarithm equations as exponent equations

Both root equations and logarithm equations can be rewritten as exponent equations.

$\sqrt{9}=3$ can be rewritten as $3^2=9$ . These two equations are the same statement about numbers, written in two different ways. $\sqrt{9}$ asks the question “What number squared is 9?” So the equation $\sqrt{9}=3$ asks this question, and then answers it: “3 squared is 9.”

We can rewrite logarithm equations in a similar way. Consider this equation:

If you are asked to rewrite that logarithm equation as an exponent equation, think about it this way. The left side asks: “3 to what power is $\left(\frac{1}{3}\right)$ ?” And the right side answers: “3 to the $-1$ power is $\left(\frac{1}{3}\right)$ .” $3^{-1}=\left(\frac{1}{3}\right)$ .

These two equations, ${\text{log}}_3\left(\frac{1}{3}\right)=-1$ and $3^{-1}=\left(\frac{1}{3}\right)$ , are two different ways of expressing the same numerical relationship.