Evaluate natural log of 4/9

Problem

ln(4/9)

Solution

1. Identify the expression as the natural logarithm of a quotient.

2. Apply the quotient rule for logarithms, which states that ln(a/b)=ln(a)−ln(b)

ln(4/9)=ln(4)−ln(9)

3. Rewrite the arguments as powers of prime numbers, noting that 4=2^2 and 9=3^2

ln(4)−ln(9)=ln(2^2)−ln(3^2)

4. Apply the power rule for logarithms, which states that ln(x^k)=k*ln(x)

ln(2^2)−ln(3^2)=2*ln(2)−2*ln(3)

5. Factor out the common constant if a simplified exact form is desired.

2*ln(2)−2*ln(3)=2*(ln(2)−ln(3))

Final Answer

ln(4/9)=2*ln(2)−2*ln(3)

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