Evaluate log base 9 of 1/5

Problem

(log_9)(1/5)

Solution

1. Apply the change of base formula to rewrite the logarithm using natural logarithms or common logarithms.

(log_b)(a)=ln(a)/ln(b)

2. Substitute the given values into the formula.

(log_9)(1/5)=ln(1/5)/ln(9)

3. Apply the quotient rule for logarithms to the numerator, noting that ln(1/5)=ln(1)−ln(5)

ln(1)=0

ln(1/5)=−ln(5)

4. Rewrite the denominator by expressing 9 as 3^2 and applying the power rule for logarithms.

ln(9)=ln(3^2)

ln(9)=2*ln(3)

5. Combine the results to find the exact simplified expression.

(log_9)(1/5)=(−ln(5))/(2*ln(3))

Final Answer

(log_9)(1/5)=−ln(5)/(2*ln(3))

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