What is ln(x)+4ln(2-y/x) ?

The solution to ln(x)+4ln(2-y/x) is ln(16x-32y+(24y^2)/x-(8y^3)/(x^2)+(y^4)/(x^3))

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simplify ln(x)+4ln(2-y/x)

simplify

ln (x) + 4 ln (2 - \frac{y}{x})

ln (16 x - 32 y + \frac{24 y ^{2}}{x} - \frac{8 y ^{3}}{x ^{2}} + \frac{y ^{4}}{x ^{3}})

Solution steps

ln (x) + 4 ln (2 - \frac{y}{x})

Apply log rule:

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

4 ln (- \frac{y}{x} + 2) = ln ((- \frac{y}{x} + 2)^{4})

= ln (x) + ln ((- \frac{y}{x} + 2)^{4})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (x) + ln ((2 - \frac{y}{x})^{4}) = ln (x (- \frac{y}{x} + 2)^{4})

= ln (x (- \frac{y}{x} + 2)^{4})

Simplify

x (2 - \frac{y}{x})^{4} : 16 x - 32 y + \frac{24 y ^{2}}{x} - \frac{8 y ^{3}}{x ^{2}} + \frac{y ^{4}}{x ^{3}}

= ln (16 x - 32 y + \frac{24 y ^{2}}{x} - \frac{8 y ^{3}}{x ^{2}} + \frac{y ^{4}}{x ^{3}})

2 lo g_{10} (x + 3) - 3 lo g_{10} (x)

ln (\frac{3 x ^{5}}{y})

- lo g_{10} (5.8 \cdot 1 0^{- 3})

ln (9) + \frac{1}{2} ln (x + 4) - 2 ln (1 + x)

ln^{3} (2 x)

What is ln(x)+4ln(2-y/x) ?
The solution to ln(x)+4ln(2-y/x) is ln(16x-32y+(24y^2)/x-(8y^3)/(x^2)+(y^4)/(x^3))