Expand the Logarithmic Expression log base 5 of 125/( square root of x+4)

Problem

(log_5)(125/√(,x+4))

Solution

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

(log_5)(125)−(log_5)(√(,x+4))

2. Rewrite the radical as a fractional exponent using the property √(n,a)=a^(1/n)

(log_5)(125)−(log_5)((x+4)^(1/2))

3. Apply the power rule for logarithms, which states that (log_b)(M^p)=p*(log_b)(M) to the second term.

(log_5)(125)−1/2*(log_5)(x+4)

4. Evaluate the constant logarithm by identifying that 125=5^3 so (log_5)(5^3)=3

3−1/2*(log_5)(x+4)

Final Answer

(log_5)(125/√(,x+4))=3−1/2*(log_5)(x+4)

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