To simplify the expression [tex]\(4 \log_3 x + 6 \log_3 y - 7 \log_3 z\)[/tex], follow these steps:
1. Apply the Power Rule of Logarithms:
The power rule of logarithms states that [tex]\( n \log_b a = \log_b (a^n) \)[/tex]. Applying this rule to each term in the expression:
[tex]\[
4 \log_3 x = \log_3 (x^4)
\][/tex]
[tex]\[
6 \log_3 y = \log_3 (y^6)
\][/tex]
[tex]\[
7 \log_3 z = \log_3 (z^7)
\][/tex]
2. Rewrite the Expression:
Substitute the simplified logarithmic terms back into the expression:
[tex]\[
\log_3 (x^4) + \log_3 (y^6) - \log_3 (z^7)
\][/tex]
3. Apply the Addition Rule of Logarithms:
The addition rule of logarithms states that [tex]\( \log_b (a) + \log_b (c) = \log_b (a \cdot c) \)[/tex]. Combine the first two logarithms:
[tex]\[
\log_3 (x^4) + \log_3 (y^6) = \log_3 (x^4 \cdot y^6)
\][/tex]
4. Apply the Subtraction Rule of Logarithms:
The subtraction rule of logarithms states that [tex]\( \log_b (a) - \log_b (c) = \log_b (a / c) \)[/tex]. Combine the results of the addition with the final term:
[tex]\[
\log_3 (x^4 \cdot y^6) - \log_3 (z^7) = \log_3 \left( \frac{x^4 \cdot y^6}{z^7} \right)
\][/tex]
Therefore, the simplified form of the expression [tex]\(4 \log_3 x + 6 \log_3 y - 7 \log_3 z\)[/tex] is:
[tex]\[
\log_3 \left( \frac{x^4 \cdot y^6}{z^7} \right)
\][/tex]
This corresponds to the given option:
[tex]\(\log_3 \left( \frac{x^4 y^6}{z^7} \right)\)[/tex]
So, the correct answer is:
[tex]\[
\boxed{\log_3 \left( \frac{x^4 y^6}{z^7} \right)}
\][/tex]
