Questions: Use the properties of logarithms (a) 3 log3 2 - log3 72 =

Solution Steps

To solve the given logarithmic expression using properties of logarithms, we can use the following steps:

1. Apply the property \( a \log_b c = \log_b (c^a) \) to the first term.
2. Use the property \( \log_b a - \log_b b = \log_b \left(\frac{a}{b}\right) \) to combine the terms.
3. Simplify the resulting expression.

We start with the expression \( 3 \log_{3} 2 - \log_{3} 72 \). Using the property \( a \log_b c = \log_b (c^a) \), we can rewrite the first term: \[ 3 \log_{3} 2 = \log_{3} (2^3) = \log_{3} 8 \]

Now, we can combine the two logarithmic terms using the property \( \log_b a - \log_b b = \log_b \left(\frac{a}{b}\right) \): \[ \log_{3} 8 - \log_{3} 72 = \log_{3} \left(\frac{8}{72}\right) \]

Next, we simplify the fraction: \[ \frac{8}{72} = \frac{1}{9} \] Thus, we have: \[ \log_{3} \left(\frac{1}{9}\right) \]

Recognizing that \( \frac{1}{9} = 3^{-2} \), we can express the logarithm as: \[ \log_{3} \left(3^{-2}\right) = -2 \]

Final Answer