What is 3t+ln(5-y)-ln(y)-2ln(2) ?

The solution to 3t+ln(5-y)-ln(y)-2ln(2) is 3t+ln((5-y)/(4y))

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simplify 3t+ln(5-y)-ln(y)-2ln(2)

simplify

3 t + ln (5 - y) - ln (y) - 2 ln (2)

3 t + ln (\frac{5 - y}{4 y})

Solution steps

3 t + ln (5 - y) - ln (y) - 2 ln (2)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (5 - y) - ln (y) = ln (\frac{- y + 5}{y})

= 3 t + ln (\frac{5 - y}{y}) - 2 ln (2)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln (2) = ln (2^{2})

= 3 t + ln (\frac{- y + 5}{y}) - ln (2^{2})

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (\frac{5 - y}{y}) - ln (2^{2}) = ln (\frac{\frac{- y + 5}{y}}{2 ^{2}})

= 3 t + ln (\frac{\frac{- y + 5}{y}}{2 ^{2}})

Simplify

\frac{\frac{5 - y}{y}}{2 ^{2}} : \frac{5 - y}{4 y}

= 3 t + ln (\frac{5 - y}{4 y})

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What is 3t+ln(5-y)-ln(y)-2ln(2) ?
The solution to 3t+ln(5-y)-ln(y)-2ln(2) is 3t+ln((5-y)/(4y))