"Expand the expression and write your answer without..." was answered by our experts

emancipezN

Answered question

2021-09-30

Expand the expression and write your answer without the exponents.

\log_{4} \frac{x^{2} y^{4}}{8}

wornoutwomanC

Skilled2021-10-01Added 81 answers

Given:

\log_{4} (\frac{x^{2} y^{4}}{8})

To find the expression of

\log_{4} (\frac{x^{2} y^{4}}{8})

Now applying the rule:

\log_{c} (\frac{a}{b}) = \log_{c} (a) - \log_{c} (b)

\log_{4} (\frac{x^{2} y^{4}}{8}) = \log_{4} (x^{2} y^{4}) - \log_{4} (8)

Now applying the rule in

\log_{4} (x^{2} y^{4})

\log_{c} (a b) = \log_{c} (a) + \log_{c} (b)

= \log_{4} (x^{2}) + \log_{4} (y^{4}) - \log_{4} (8)

Now Applying the rule (3) in

\log_{4} (x^{2}) + \log_{4} (y^{4})

= 2 \log_{4} (x) + 4 \log_{4} (y) - \log_{4} (8)

= 2 \log_{4} (x) + 4 \log_{4} (y) - \log_{2^{2}} (8)

Now Applying the rule (3) in

\log_{2} (8)

\log_{a} (x^{b}) = b \log_{a} (x)

= 2 \log_{4} (x) + 4 \log_{4} (y) - \frac{1}{2} 3 \log_{2} (2)

Now Applying the rule (5) in

\log_{2} (2)

\log_{a} (a) = 1

= 2 \log_{4} (x) + 41 \log_{4} (y) - \frac{3}{2}

Answer:

\log_{4} (\frac{x^{2} y^{4}}{8}) = 2 \log_{4} (x) + 4 \log_{4} (y) = \frac{3}{2}

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