Resolved: "Expand using logarithmic properties. Where possible, evaluate logarithmic..."

Dolly Robinson

Answered question

2021-08-17

Expand using logarithmic properties. Where possible, evaluate logarithmic expressions.

\log_{5} (\frac{x^{3} \sqrt{y}}{125})

Answer & Explanation

Alix Ortiz

Skilled2021-08-18Added 109 answers

Step 1
On simplification, we get

\log_{5} (\frac{x^{3} \sqrt{y}}{125}) = \log_{5} (125) [∵ \log_{a} (\frac{m}{n}) = \log_{a} (m) - \log_{a} (n)]

= \log_{5} (x^{3}) + \log_{5} (\sqrt{y}) - \log_{5} (5^{3}) [∵ \log_{a} (m n) = \log_{a} (m) + \log_{a} (n)]

= \log_{5} (x^{3}) + \log_{5} (y^{\frac{1}{2}}) - \log_{5} (5^{3})

= 3 \log_{5} (x) + \frac{1}{2} \log_{5} (y) - 3 \log_{5} (5) [∵ \log_{a} (m^{n}) = n \log_{a} (m)]

= 3 \log_{5} (x) + \frac{1}{2} \log_{5} (y) - 3 (1) [∵ \log_{a} (a) = 1]

= 3 \log_{5} (x) + \frac{1}{2} \log_{5} (y) - 3

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