Question: (\[ 30^{-2} = \frac{1}{900} \] \[ \log_{30} \frac{1}{900} = -2 \] \[ \log_{-2}…

\(\[ 30^{-2} = \frac{1}{900} \]

\[ \log_{30} \frac{1}{900} = -2 \]

\[ \log_{-2} \frac{1}{900} = 30 \]

\[ \log_{\frac{1}{900}} 30 = -2 \]

\[ \log_{30} -2 = \frac{1}{900} \]\)

Solution

We are given four logarithmic expressions and need to determine which one is correct based on \(30^{-2} = \frac{1}{900}\). Let’s verify the first option: 1. \(\log_{30} \frac{1}{900} = -2\) This implies: \[ 30^{-2} = \frac{1}{900} \] This is a true statement. Therefore, this option is correct. Let’s quickly check if any other options are correct: 2. \(\log_{-2} \frac{1}{900} = 30\) Here, a logarithm with a negative base is not defined in real numbers, so this cannot be correct. 3. \(\log_{\frac{1}{900}} 30 = -2\) This implies: \[ \left(\frac{1}{900}\right)^{-2} = 30 \] This statement is not true, so this is incorrect. 4. \(\log_{30} - 2 = \frac{1}{900}\) There is no meaningful interpretation of this expression as written in terms of logarithmic rules. Thus, the correct option is: \(\log_{30} \frac{1}{900} = -2\)