What is (-ln(a/P-b)+ln(a/(P_{0)}-b))/a ?

The solution to (-ln(a/P-b)+ln(a/(P_{0)}-b))/a is (ln((P(a-bP_{0}))/(P_{0)(a-bP)}))/a

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simplify (-ln(a/P-b)+ln(a/(P_{0)}-b))/a

simplify

\frac{- ln ( \frac{a}{P} - b ) + ln ( \frac{a}{P _{0}} - b )}{a}

\frac{ln ( \frac{P ( a - b P _{0} )}{P _{0} ( a - b P )} )}{a}

Solution steps

\frac{- ln ( \frac{a}{P} - b ) + ln ( \frac{a}{P _{0}} - b )}{a}

Apply log rule:

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

ln (\frac{a}{P _{0}} - b) - ln (\frac{a}{P} - b) = ln (\frac{\frac{a}{P _{0}} - b}{\frac{a}{P} - b})

= \frac{ln ( \frac{\frac{a}{P _{0}} - b}{\frac{a}{P} - b} )}{a}

Simplify

\frac{\frac{a}{P _{0}} - b}{\frac{a}{P} - b} : \frac{P ( a - b P _{0} )}{P _{0} ( a - b P )}

= \frac{ln ( \frac{P ( a - b P _{0} )}{P _{0} ( a - b P )} )}{a}

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What is (-ln(a/P-b)+ln(a/(P_{0)}-b))/a ?
The solution to (-ln(a/P-b)+ln(a/(P_{0)}-b))/a is (ln((P(a-bP_{0}))/(P_{0)(a-bP)}))/a