Evaluate log base 9 of 15/7

Problem

(log_9)(15/7)

Solution

1. Apply the quotient rule for logarithms, which states that (log_b)(M/N)=(log_b)(M)−(log_b)(N)

(log_9)(15/7)=(log_9)(15)−(log_9)(7)

2. Apply the change of base formula to convert the expression into natural logarithms or common logarithms using (log_b)(a)=ln(a)/ln(b)

(log_9)(15)−(log_9)(7)=ln(15)/ln(9)−ln(7)/ln(9)

3. Combine the fractions since they share a common denominator.

(ln(15)−ln(7))/ln(9)

4. Simplify the denominator by recognizing that 9=3 and using the power rule ln(xp)=p*ln(x)

ln(9)=ln(3)=2*ln(3)

5. Substitute the simplified denominator back into the expression.

(ln(15)−ln(7))/(2*ln(3))

Final Answer

(log_9)(15/7)=(ln(15)−ln(7))/(2*ln(3))

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