Questions: Suppose we want to combine the following into a single expression: 6 log3(x) + (1/0) log3(y) - 5 log3(z)

Solution Steps

To combine the given logarithmic expressions into a single expression, we will use the power rule of logarithms, which states that \( a \log_b(c) = \log_b(c^a) \). We will apply this rule to each term and then combine them using the properties of logarithms: the product rule \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\) and the quotient rule \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\).

1. Apply the power rule to each term:
   • \(6 \log_3(x) = \log_3(x^6)\)
   • \(\frac{1}{0} \log_3(y)\) is undefined due to division by zero, so this term cannot be combined.
   • \(-5 \log_3(z) = \log_3(z^{-5})\)
2. Combine the valid terms using the product and quotient rules.

We start with the expression: \[ 6 \log_{3}(x) + \frac{1}{0} \log_{3}(y) - 5 \log_{3}(z) \] Applying the power rule \(a \log_b(c) = \log_b(c^a)\) to the valid terms, we have: \[ 6 \log_{3}(x) = \log_{3}(x^6) \] \[ -5 \log_{3}(z) = \log_{3}(z^{-5}) \] The term \(\frac{1}{0} \log_{3}(y)\) is undefined and will be ignored.

Now we can combine the valid logarithmic terms using the product and quotient rules: \[ \log_{3}(x^6) - \log_{3}(z^5) = \log_{3}\left(\frac{x^6}{z^5}\right) \]

Substituting \(x = 2\) and \(z = 3\) into the expression: \[ \log_{3}\left(\frac{2^6}{3^5}\right) \] Calculating \(2^6 = 64\) and \(3^5 = 243\), we have: \[ \log_{3}\left(\frac{64}{243}\right) \]

The numerical value of the logarithm is calculated as: \[ \log_{3}\left(\frac{64}{243}\right) \approx -1.2144 \]

Final Answer