Expand the Logarithmic Expression log base 2 of fourth root of 8

Problem

(log_2)(√(4,8))

Solution

1. Rewrite the radical expression using a fractional exponent.

√(4,8)=8(1/4)

2. Apply the power rule for logarithms, which states (log_b)(xp)=p*(log_b)(x)

(log_2)(8(1/4))=1/4*(log_2)(8)

3. Identify that the base of the logarithm and the argument are both powers of 2.

8=2

4. Substitute the power of 2 back into the expression.

1/4*(log_2)(2)

5. Simplify the logarithm using the property (log_b)(bx)=x

(log_2)(2)=3

6. Multiply the resulting values to find the final expanded and simplified form.

1/4⋅3=3/4

Final Answer

(log_2)(√(4,8))=3/4

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