To simplify the expression [tex]\(4 \log_3 y - 6 \log_3 x + 7 \log_3 z\)[/tex], we will use the properties of logarithms, specifically the power rule and the properties of logarithms involving sums and differences.
1. Apply the Power Rule: The power rule of logarithms states that [tex]\(a \log_b c = \log_b (c^a)\)[/tex]. Applying this to each term in the expression, we get:
[tex]\[
4 \log_3 y = \log_3 (y^4)
\][/tex]
[tex]\[
6 \log_3 x = \log_3 (x^6)
\][/tex]
[tex]\[
7 \log_3 z = \log_3 (z^7)
\][/tex]
So the expression now becomes:
[tex]\[
\log_3 (y^4) - \log_3 (x^6) + \log_3 (z^7)
\][/tex]
2. Combine Logarithms Using the Difference and Sum Rules: The difference rule for logarithms states that [tex]\(\log_b (a) - \log_b (b) = \log_b \left(\frac{a}{b}\right)\)[/tex]. Combining the first two terms, we have:
[tex]\[
\log_3 (y^4) - \log_3 (x^6) = \log_3 \left(\frac{y^4}{x^6}\right)
\][/tex]
Now our expression looks like:
[tex]\[
\log_3 \left(\frac{y^4}{x^6}\right) + \log_3 (z^7)
\][/tex]
The sum rule for logarithms states that [tex]\(\log_b (a) + \log_b (b) = \log_b (ab)\)[/tex]. Combining these two terms, we get:
[tex]\[
\log_3 \left(\frac{y^4}{x^6}\right) + \log_3 (z^7) = \log_3 \left(\frac{y^4}{x^6} \cdot z^7\right)
\][/tex]
3. Write the Final Simplified Form: Therefore, the simplified form of the given expression is:
[tex]\[
\log_3 \left(\frac{y^4 z^7}{x^6}\right)
\][/tex]
Thus, the simplified form of [tex]\(4 \log_3 y - 6 \log_3 x + 7 \log_3 z\)[/tex] is:
[tex]\[
\log_3 \left(\frac{y^4 z^7}{x^6}\right)
\][/tex]
