1. Rewrite the expression 1/2ln(x – 6) + 3ln(x +1) as a single logarithm.
2. Rewrite the expression 1/3[ln x – 2ln(1 – x)] as a single logarithm.
#1: ln[√(x-6)*(x+1)^3]
#2: ln∛(x/(1-x)^2)
To rewrite the expressions as a single logarithm, we need to use the properties of logarithms. Two properties that are particularly useful for this task are the product rule and the quotient rule.
1. To rewrite the expression 1/2ln(x – 6) + 3ln(x +1) as a single logarithm, we can use the product rule. The product rule states that ln(a) + ln(b) = ln(ab). Applying this rule, we have:
1/2ln(x – 6) + 3ln(x +1) = ln((x - 6)^(1/2)) + ln((x + 1)^3).
Now, we can use the product rule again to combine the two logarithms into a single logarithm:
ln((x - 6)^(1/2)) + ln((x + 1)^3) = ln( (x - 6)^(1/2) * (x + 1)^3 ).
So, the expression 1/2ln(x – 6) + 3ln(x +1) can be rewritten as ln( (x - 6)^(1/2) * (x + 1)^3 ).
2. To rewrite the expression 1/3[ln x – 2ln(1 – x)] as a single logarithm, we can use the quotient rule and the power rule. The quotient rule states that ln(a) - ln(b) = ln(a/b), and the power rule states that ln(a^n) = nln(a).
Applying the quotient rule, we have:
1/3[ln x – 2ln(1 – x)] = 1/3ln(x / (1 - x^2)).
Now, we can simplify further using the power rule:
1/3ln(x / (1 - x^2)) = ln((x / (1 - x^2))^(1/3)).
Therefore, the expression 1/3[ln x – 2ln(1 – x)] can be rewritten as ln((x / (1 - x^2))^(1/3)).