Use the Laws of Logarithms to combine the expression.
1/5log(x + 3)^5 +1/3 [log x^6 − log(x^2 − x − 12)^3]
1/5 log(x+3)^5 = log(x+3)
log(x^6)-log(x^2-x-12)^3
= 3log(x^2/(x^2-x-12)^3)
= log((x^2)/(x^2-x-12))
so, adding those logs, we have
log(x+3) + log((x^2)/(x^2-x-12))
= log(x^2(x+3) / (x+3)(x-4))
= log(x^2/(x-4))
To combine the given expression using the laws of logarithms, we first need to apply the properties of logarithms.
The first property we'll use is the power rule, which states that log_b(x^n) = n*log_b(x). Using this property, we can rewrite the first term as:
1/5 * log((x + 3)^5) = log((x + 3)^5)^(1/5) = log(x + 3)
The second property we'll use is the quotient rule, which states that log_b(x/y) = log_b(x) - log_b(y). Using this property, we can rewrite the second term as:
1/3 * [log(x^6) - log((x^2 - x - 12)^3)]
Now let's simplify the expression:
1/3 * [log(x^6) - log((x^2 - x - 12)^3)]
= 1/3 * (log(x^6) - 3*log(x^2 - x - 12))
Next, we'll use the power rule to simplify the first logarithm:
1/3 * (6*log(x) - 3*log(x^2 - x - 12))
Now, we can distribute the 1/3 to both terms:
(1/3) * 6*log(x) - (1/3) * 3*log(x^2 - x - 12)
Simplifying further, we get:
2*log(x) - log(x^2 - x - 12)
Therefore, the combined expression is 2*log(x) - log(x^2 - x - 12).