Expand the Logarithmic Expression log base 6 of (z/36)^4

Problem

(log_6)(z/36)

Solution

1. Apply the power rule for logarithms, which states that (log_b)(Mp)=p*(log_b)(M) by moving the exponent 4 to the front of the expression.

(log_6)(z/36)=4*(log_6)(z/36)

2. Apply the quotient rule for logarithms, which states that (log_b)(M/N)=(log_b)(M)−(log_b)(N) to the term inside the parentheses.

4*(log_6)(z/36)=4*((log_6)(z)−(log_6)(36))

3. Evaluate the constant logarithm by determining the power to which the base 6 must be raised to get 36 Since 6=36 it follows that (log_6)(36)=2

4*((log_6)(z)−(log_6)(36))=4*((log_6)(z)−2)

4. Distribute the constant 4 to both terms inside the parentheses to reach the final expanded form.

4*((log_6)(z)−2)=4*(log_6)(z)−8

Final Answer

(log_6)(z/36)=4*(log_6)(z)−8

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