Expand the Logarithmic Expression log base 9 of square root of 3

Problem

(log_9)(√(,3))

Solution

1. Rewrite the square root using a fractional exponent.

√(,3)=3(1/2)

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

(log_9)(3(1/2))=1/2*(log_9)(3)

3. Evaluate the remaining logarithm by expressing the base 9 as a power of 3

9=3

4. Determine the value of (log_9)(3) by solving 9=3

(3)y=3

3(2*y)=3

2*y=1

y=1/2

5. Substitute the value back into the expression.

1/2⋅1/2=1/4

Final Answer

(log_9)(√(,3))=1/4

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