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