What is ln(x/y)-2ln(x^3)-4ln(y) ?

The solution to ln(x/y)-2ln(x^3)-4ln(y) is -ln(x^5y^5)

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simplify ln(x/y)-2ln(x^3)-4ln(y)

simplify

ln (\frac{x}{y}) - 2 ln (x^{3}) - 4 ln (y)

- ln (x^{5} y^{5})

Solution steps

ln (\frac{x}{y}) - 2 ln (x^{3}) - 4 ln (y)

Apply log rule:

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

2 ln (x^{3}) = ln ((x^{3})^{2})

= ln (\frac{x}{y}) - ln ((x^{3})^{2}) - 4 ln (y)

Apply log rule:

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

ln (\frac{x}{y}) - ln ((x^{3})^{2}) = ln (\frac{\frac{x}{y}}{( x ^{3} ) ^{2}})

= ln (\frac{\frac{x}{y}}{( x ^{3} ) ^{2}}) - 4 ln (y)

Simplify

\frac{\frac{x}{y}}{( x ^{3} ) ^{2}} : \frac{1}{x ^{5} y}

= ln (\frac{1}{x ^{5} y}) - 4 ln (y)

Apply log rule:

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

4 ln (y) = ln (y^{4})

= ln (\frac{1}{x ^{5} y}) - ln (y^{4})

Apply log rule:

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

ln (\frac{1}{x ^{5} y}) - ln (y^{4}) = ln (\frac{\frac{1}{x ^{5} y}}{y ^{4}})

= ln (\frac{\frac{1}{x ^{5} y}}{y ^{4}})

Simplify

\frac{\frac{1}{x ^{5} y}}{y ^{4}} : \frac{1}{x ^{5} y ^{5}}

= ln (\frac{1}{x ^{5} y ^{5}})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (x^{5} y^{5})

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

x^{5} y^{5} = (x y)^{5}

= - ln ((x y)^{5})

(x y)^{5} = x^{5} y^{5}

= - ln (x^{5} y^{5})

ln (\frac{e ^{x l n (3)}}{3})

5 lo g_{b} (x) + 6 lo g_{b} (y)

\frac{1}{5} (lo g_{3} (x) + lo g_{3} (x - 2))

lo g_{10} (5) + lo g_{10} (3)

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