Final answer:
To combine the given expression using the laws of logarithms, we apply the property that allows exponents to be brought in front of the logarithm and the properties that define the logarithm of a product and quotient. This simplifies the expression to a single logarithm representing a complex rational expression.
Step-by-step explanation:
Using the laws of logarithms, we can combine the expression (1/5)log(x+3)⁵ + (1/3)[log(x⁶) - log(x²-x-12)³].
- First, we use the third logarithm property that allows us to bring the exponent in front of the logarithm: log(a^n) = n · log(a). This property applies to both terms in the expression. So, we can rewrite the expression as:
log((x+3)) + 1/3 · (log(x⁶) - log((x²-x-12)³)). - Next, we use the first logarithm property, which states that the logarithm of a product is the sum of the logarithms: log(a · b) = log(a) + log(b), and the logarithm of a quotient is the difference of the logarithms: log(a / b) = log(a) - log(b). Hence, we can combine the terms within the brackets to get:
1/3 · (log(x⁶ / (x²-x-12)³)). - Finally, we apply the first property again to the term 1/3 log(x⁶ / (x²-x-12)³), which allows us to write it as:
log((x⁶ / (x²-x-12)³)¹⁴³). Combining everything gives us a single logarithm expression:
log((x+3) · (x⁶ / (x²-x-12)¹⁴³)).