Rewrite the logarithmic equation as an exponential equation.
${\log }_3\left(\frac{1}{27}\right)=-3$

• A. $-3^3=\frac{1}{27}$
• B. $-3^3=-27$
• C. $3^{-3}=\frac{1}{27}$
• D. $3^{-3}=-\frac{1}{27}$

Answer: (C)

Verified step by step guidance

1. Recall the definition of logarithms: if \(\log_b(x) = y\), then the equivalent exponential form is \(b^y = x\).
2. Identify the base \(b\), the logarithm result \(y\), and the argument \(x\) from the given equation \(\log_3\left(\frac{1}{27}\right) = -3\).
3. Here, the base is 3, the logarithm result is \(-3\), and the argument is \(\frac{1}{27}\).
4. Rewrite the logarithmic equation as an exponential equation by raising the base 3 to the power of \(-3\) and setting it equal to \(\frac{1}{27}\): \$3^{-3} = \frac{1}{27}$.
5. This shows the equivalence between the logarithmic and exponential forms without calculating the numerical value.