Use the properties of logarithms to condense the expression
1/3[ln(x - 6)- 4 ln y - 2 ln z]
ln [(x-6)/(y^4z^2)]^1/3
To condense the expression using the properties of logarithms, we can apply the following rules:
1. Product rule: log_a(b) + log_a(c) = log_a(b * c)
2. Quotient rule: log_a(b) - log_a(c) = log_a(b / c)
3. Power rule: log_a(b^c) = c * log_a(b)
Let's apply these rules step by step to condense the expression:
1/3[ln(x - 6)- 4 ln y - 2 ln z]
First, we can use the power rule to simplify the coefficients of the logarithms:
1/3[ln(x - 6) + ln(y^(-4)) + ln(z^(-2))]
Next, we can use the product rule to combine the three logarithms into a single logarithm:
1/3[ln((x - 6) * y^(-4) * z^(-2))]
Then, we can simplify further by applying the power rule to each term inside the logarithm:
1/3[ln((x - 6) / (y^4 * z^2))]
Finally, we can remove the brackets to obtain the condensed form:
ln((x - 6) / (y^4 * z^2)) / 3
Therefore, the condensed form of the expression is ln((x - 6) / (y^4 * z^2)) / 3.