Expand the Logarithmic Expression log of square root of b^7c

Problem

(log_)(√(,b7*c))

Solution

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

(log_)((b7*c)(1/2))

2. Apply the power rule for logarithms, (log_)(xn)=n*(log_)(x) to move the exponent to the front.

1/2*(log_)(b7*c)

3. Apply the product rule for logarithms, (log_)(x*y)=(log_)(x)+(log_)(y) to separate the terms inside the parentheses.

1/2*((log_)(b7)+(log_)(c))

4. Apply the power rule again to the term (log_)(b7) to move the exponent 7 to the front.

1/2*(7*(log_)(b)+(log_)(c))

5. Distribute the fraction 1/2 to both terms to reach the final expanded form.

7/2*(log_)(b)+1/2*(log_)(c)

Final Answer

(log_)(√(,b7*c))=7/2*(log_)(b)+1/2*(log_)(c)

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