Evaluate log base 3 of 1/18

Problem

(log_3)(1/18)

Solution

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

(log_3)(1/18)=(log_3)(1)−(log_3)(18)

2. Evaluate the first term using the property that (log_b)(1)=0 for any base b

(log_3)(1)=0

3. Factor the argument of the second term to identify powers of the base.

18=9⋅2=3⋅2

4. Apply the product rule for logarithms, which states that (log_b)(x*y)=(log_b)(x)+(log_b)(y)

(log_3)(18)=(log_3)(3⋅2)

(log_3)(18)=(log_3)(3)+(log_3)(2)

5. Simplify the power using the property (log_b)(bk)=k

(log_3)(3)=2

(log_3)(18)=2+(log_3)(2)

6. Combine the results back into the original expression.

(log_3)(1/18)=0−(2+(log_3)(2))

(log_3)(1/18)=−2−(log_3)(2)

Final Answer

(log_3)(1/18)=−2−(log_3)(2)

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