Rewrite the exponential equation as a logarithmic equation.
$6^{-3}=\frac{1}{216}$

• A. ${\log }_{6}216=-3$
• B. ${\log }_{-3}\frac{1}{216}=6$
• C. ${\log }_6\frac{1}{216}=-3$
• D. ${\log }_{216}6=-3$

Answer: (C)

Verified step by step guidance

1. Recall the relationship between exponential and logarithmic forms: if \(a^{b} = c\), then the equivalent logarithmic form is \(\log_{a} c = b\).
2. Identify the base \(a\), the exponent \(b\), and the result \(c\) from the given exponential equation \$6^{-3} = \frac{1}{216}$.
3. Here, the base \(a\) is 6, the exponent \(b\) is \(-3\), and the result \(c\) is \(\frac{1}{216}\).
4. Rewrite the equation in logarithmic form by placing the base 6 as the base of the logarithm, the result \(\frac{1}{216}\) as the argument of the logarithm, and set it equal to the exponent \(-3\).
5. The logarithmic form is therefore \(\log_{6} \frac{1}{216} = -3\).