What is ln(xy)+7ln(y/z)-5ln(xyz) ?

The solution to ln(xy)+7ln(y/z)-5ln(xyz) is ln((y^3)/(x^4z^{12)})

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simplify ln(xy)+7ln(y/z)-5ln(xyz)

simplify

ln (x y) + 7 ln (\frac{y}{z}) - 5 ln (x yz)

ln (\frac{y ^{3}}{x ^{4} z ^{12}})

Solution steps

ln (x y) + 7 ln (\frac{y}{z}) - 5 ln (x yz)

Apply log rule:

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

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

= ln (x y) + ln ((\frac{y}{z})^{7}) - 5 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})^{7}) = ln ((\frac{y}{z})^{7} x y)

= ln ((\frac{y}{z})^{7} x y) - 5 ln (x yz)

Simplify

x y (\frac{y}{z})^{7} : \frac{x y ^{8}}{z ^{7}}

= ln (\frac{x y ^{8}}{z ^{7}}) - 5 ln (x yz)

Apply log rule:

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

5 ln (x yz) = ln ((x yz)^{5})

= ln (\frac{x y ^{8}}{z ^{7}}) - ln ((x yz)^{5})

Apply log rule:

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

ln (\frac{x y ^{8}}{z ^{7}}) - ln ((x yz)^{5}) = ln (\frac{\frac{x y ^{8}}{z ^{7}}}{( x yz ) ^{5}})

= ln (\frac{\frac{x y ^{8}}{z ^{7}}}{( x yz ) ^{5}})

Simplify

\frac{\frac{x y ^{8}}{z ^{7}}}{( x yz ) ^{5}} : \frac{y ^{3}}{x ^{4} z ^{12}}

= ln (\frac{y ^{3}}{x ^{4} z ^{12}})

lo g_{4} (3.188 - 1) + lo g_{4} (2 \cdot 3.188 - 5)

ln (\frac{e ^{2} x ^{2} - 3}{x ( x + 1 ) ^{5}})

\frac{1}{3} ln (x) + \frac{1}{4} ln (y) + c

2 ln (e) + 2 lo g_{x} (+ 3 lo g_{y} (x))

ln (\frac{1}{π})

What is ln(xy)+7ln(y/z)-5ln(xyz) ?
The solution to ln(xy)+7ln(y/z)-5ln(xyz) is ln((y^3)/(x^4z^{12)})