Final answer:
The logarithmic expression can be condensed using the properties of logarithms: the product rule and quotient rule. The condensed expression is ln{[((x+9)^6 × x) / (x^6 - 8)]^(1/2)}.
Step-by-step explanation:
The student's question involves condensing the given logarithmic expression into a single logarithm. To achieve this, we apply the laws of logarithms. Utilizing these properties allows us to manipulate and simplify the expression accordingly.
The two key properties of logarithms needed here are:
- Logarithm of a product: The logarithm of a product of two numbers is the sum of the logarithms of the two numbers which can be expressed as ln(xy) = ln(x) + ln(y).
- Logarithm of a quotient: The logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers, written as ln(x/y) = ln(x) - ln(y).
To condense the given expression (1)/(2)[6ln(x+9)+ln(x)-ln(x6-8)], we first apply the logarithm of a product rule to the terms 6ln(x+9) and ln(x), which gives us ln((x+9)6 × x). Then, we apply the logarithm of a quotient rule to this result and the term ln(x6 - 8), leading to ln{((x+9)6 × x) / (x6 - 8)}. Finally, the factor of 1/2 can be brought into the logarithm as an exponent, yielding the condensed logarithmic expression ln{[((x+9)6 × x) / (x6 - 8)]1/2}.