Expand the Logarithmic Expression log base 3 of 405/x

Problem

(log_3)(405/x)

Solution

1. Identify the logarithmic property for division, which is the Quotient Rule: (log_b)(M/N)=(log_b)(M)−(log_b)(N)

2. Apply the Quotient Rule to the expression to separate the numerator and the denominator.

(log_3)(405)−(log_3)(x)

3. Factor the number 405 to see if it can be simplified using base 3

405=81⋅5=3⋅5

4. Apply the Product Rule (log_b)(M*N)=(log_b)(M)+(log_b)(N) to the first term.

(log_3)(3⋅5)−(log_3)(x)

(log_3)(3)+(log_3)(5)−(log_3)(x)

5. Simplify the term (log_3)(3) using the property (log_b)(by)=y

4+(log_3)(5)−(log_3)(x)

Final Answer

(log_3)(405/x)=4+(log_3)(5)−(log_3)(x)

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