Simplify/Condense log base 9 of 25- log base 9 of 75

Problem

(log_9)(25)−(log_9)(75)

Solution

1. Identify the logarithmic property for subtraction, which is the quotient rule: (log_b)(M)−(log_b)(N)=(log_b)(M/N)

2. Apply the rule to the given expression by placing the arguments into a single logarithm as a fraction.

(log_9)(25/75)

3. Simplify the fraction inside the logarithm by dividing both the numerator and the denominator by their greatest common factor, 25.

(log_9)(1/3)

4. Rewrite the argument as a power of 3 to prepare for further simplification.

(log_9)(3^(−1))

5. Change the base of the logarithm or express the base 9 as 3^2 to evaluate the expression.

(log_3^2)(3^(−1))

6. Apply the property (log_b^k)(x^p)=p/k*(log_b)(x) to find the numerical value.

(−1)/2*(log_3)(3)

7. Evaluate the remaining logarithm since (log_3)(3)=1

−1/2

Final Answer

(log_9)(25)−(log_9)(75)=−1/2

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