Evaluate

Problem

(log_3)(100)−(log_3)(18)−(log_3)(50)

Solution

1. Identify the logarithmic property for subtraction, which states that (log_b)(M)−(log_b)(N)=(log_b)(M/N)

2. Apply the property to the first two terms of the expression.

(log_3)(100)−(log_3)(18)=(log_3)(100/18)

3. Simplify the fraction inside the logarithm by dividing the numerator and denominator by 2.

(log_3)(100/18)=(log_3)(50/9)

4. Apply the property again to subtract the final term from the result.

(log_3)(50/9)−(log_3)(50)=(log_3)(50/9/50)

5. Simplify the complex fraction by multiplying by the reciprocal.

50/9/50=50/9⋅1/50

50/9⋅1/50=1/9

6. Evaluate the resulting logarithm by expressing the argument as a power of the base 3.

(log_3)(1/9)=(log_3)(3(−2))

(log_3)(3(−2))=−2

Final Answer

(log_3)(100)−(log_3)(18)−(log_3)(50)=−2

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