What is ln(xy)+9ln(y/z)-8ln(xyz) ?

The solution to ln(xy)+9ln(y/z)-8ln(xyz) is ln((y^2)/(x^7z^{17)})

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simplify ln(xy)+9ln(y/z)-8ln(xyz)

simplify

ln (x y) + 9 ln (\frac{y}{z}) - 8 ln (x yz)

ln (\frac{y ^{2}}{x ^{7} z ^{17}})

Solution steps

ln (x y) + 9 ln (\frac{y}{z}) - 8 ln (x yz)

Apply log rule:

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

9 ln (\frac{y}{z}) = ln ((\frac{y}{z})^{9})

= ln (x y) + ln ((\frac{y}{z})^{9}) - 8 ln (x yz)

Apply log rule:

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

ln (x y) + ln ((\frac{y}{z})^{9}) = ln ((\frac{y}{z})^{9} x y)

= ln ((\frac{y}{z})^{9} x y) - 8 ln (x yz)

Simplify

x y (\frac{y}{z})^{9} : \frac{x y ^{10}}{z ^{9}}

= ln (\frac{x y ^{10}}{z ^{9}}) - 8 ln (x yz)

Apply log rule:

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

8 ln (x yz) = ln ((x yz)^{8})

= ln (\frac{x y ^{10}}{z ^{9}}) - ln ((x yz)^{8})

Apply log rule:

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

ln (\frac{x y ^{10}}{z ^{9}}) - ln ((x yz)^{8}) = ln (\frac{\frac{x y ^{10}}{z ^{9}}}{( x yz ) ^{8}})

= ln (\frac{\frac{x y ^{10}}{z ^{9}}}{( x yz ) ^{8}})

Simplify

\frac{\frac{x y ^{10}}{z ^{9}}}{( x yz ) ^{8}} : \frac{y ^{2}}{x ^{7} z ^{17}}

= ln (\frac{y ^{2}}{x ^{7} z ^{17}})

5 lo g_{10} (y) - 3 lo g_{10} (m)

5 lo g_{6} (u) + 8 lo g_{6} (v)

- lo g_{10} (4 \cdot 1 0^{- 8}) + lo g_{10} (1)

2.338 - 1.5 - 0.287 ln (\frac{675}{45})

lo g_{3} (\frac{( x + 4 ) ^{2}}{y ^{3} z})

What is ln(xy)+9ln(y/z)-8ln(xyz) ?
The solution to ln(xy)+9ln(y/z)-8ln(xyz) is ln((y^2)/(x^7z^{17)})