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. Since the problem is titled 'Intro to Logarithms Practice 5', let's start by recalling the basic definition of a logarithm: for \(a > 0\), \(a \neq 1\), and \(x > 0\), the logarithm \(\log_a x = y\) means that \(a^y = x\).
2. Identify the specific logarithmic expression or equation given in the problem (for example, something like \(\log_a b = c\)) and write down what it means in exponential form using the definition: \(a^c = b\).
3. If the problem involves solving for a variable inside the logarithm, isolate the logarithmic expression first, then rewrite it in exponential form to solve for the variable.
4. Use logarithm properties if needed, such as the product rule \(\log_a (xy) = \log_a x + \log_a y\), the quotient rule \(\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y\), or the power rule \(\log_a (x^k) = k \log_a x\), to simplify the expression before solving.
5. After rewriting and simplifying, solve the resulting equation for the unknown variable, making sure to check that your solution is valid within the domain of the logarithm (i.e., the argument of the logarithm must be positive).