Expand the Logarithmic Expression log base 4 of square root of 5x

Problem

(log_4)(√(,5*x))

Solution

1. Rewrite the square root as a fractional exponent using the property √(,a)=a(1/2)

(log_4)((5*x)(1/2))

2. Apply the power rule for logarithms, (log_b)(Mp)=p*(log_b)(M) to move the exponent to the front.

1/2*(log_4)(5*x)

3. Apply the product rule for logarithms, (log_b)(M*N)=(log_b)(M)+(log_b)(N) to separate the terms inside the logarithm.

1/2*((log_4)(5)+(log_4)(x))

4. Distribute the constant factor to each term in the expression.

1/2*(log_4)(5)+1/2*(log_4)(x)

Final Answer

(log_4)(√(,5*x))=1/2*(log_4)(5)+1/2*(log_4)(x)

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