Simplify/Condense

Problem

ln(3)+1/3*ln(4−x2)−ln(x)

Solution

1. Apply the power rule for logarithms to the middle term, which states a*ln(b)=ln(ba)

1/3*ln(4−x2)=ln((4−x2)(1/3))

2. Apply the product rule for logarithms to the first two terms, which states ln(a)+ln(b)=ln(a*b)

ln(3)+ln((4−x2)(1/3))=ln(3*(4−x2)(1/3))

3. Apply the quotient rule for logarithms to the remaining terms, which states ln(a)−ln(b)=ln(a/b)

ln(3*(4−x2)(1/3))−ln(x)=ln((3*(4−x2)(1/3))/x)

4. Rewrite the exponent as a radical for the final simplified form.

(4−x2)(1/3)=√(3,4−x2)

Final Answer

ln(3)+1/3*ln(4−x2)−ln(x)=ln((3√(3,4−x2))/x)

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