Expand the Logarithmic Expression log base 5 of (z/25)^3

Problem

(log_5)(z/25)

Solution

1. Apply the power property of logarithms, which states that (log_b)(Mp)=p*(log_b)(M) to move the exponent in front of the logarithm.

(log_5)(z/25)=3*(log_5)(z/25)

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

3*(log_5)(z/25)=3*((log_5)(z)−(log_5)(25))

3. Evaluate the constant logarithm by determining the power to which the base 5 must be raised to get 25

(log_5)(25)=2

4. Substitute and distribute the constant 3 to both terms inside the parentheses to reach the final expanded form.

3*((log_5)(z)−2)=3*(log_5)(z)−6

Final Answer

(log_5)(z/25)=3*(log_5)(z)−6

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