Expand the Logarithmic Expression log base 2 of (a^2 square root of b)^4

Problem

(log_2)(a2√(,b))

Solution

1. Apply the power rule for logarithms, which states (log_b)(xn)=n*(log_b)(x) to move the outer exponent of 4 to the front of the expression.

(log_2)(a2√(,b))=4*(log_2)(a2√(,b))

2. Apply the product rule for logarithms, (log_b)(x*y)=(log_b)(x)+(log_b)(y) to separate the terms inside the logarithm.

4*(log_2)(a2√(,b))=4*((log_2)(a2)+(log_2)(√(,b)))

3. Rewrite the radical as a fractional exponent, where √(,b)=b(1/2)

4*((log_2)(a2)+(log_2)(√(,b)))=4*((log_2)(a2)+(log_2)(b(1/2)))

4. Apply the power rule again to the individual terms inside the parentheses to move the exponents 2 and 1/2 to the front of their respective logarithms.

4*((log_2)(a2)+(log_2)(b(1/2)))=4*(2*(log_2)(a)+1/2*(log_2)(b))

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

4*(2*(log_2)(a)+1/2*(log_2)(b))=8*(log_2)(a)+2*(log_2)(b)

Final Answer

(log_2)(a2√(,b))=8*(log_2)(a)+2*(log_2)(b)

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