What is ln(x)-ln(y)-6ln(x)-4ln(y) ?

The solution to ln(x)-ln(y)-6ln(x)-4ln(y) is -ln(x^5y^5)

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simplify ln(x)-ln(y)-6ln(x)-4ln(y)

simplify

ln (x) - ln (y) - 6 ln (x) - 4 ln (y)

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

Solution steps

ln (x) - ln (y) - 6 ln (x) - 4 ln (y)

Apply log rule:

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

ln (x) - ln (y) = ln (\frac{x}{y})

= ln (\frac{x}{y}) - 6 ln (x) - 4 ln (y)

Apply log rule:

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

6 ln (x) = ln (x^{6})

= ln (\frac{x}{y}) - ln (x^{6}) - 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^{6}) = ln (\frac{\frac{x}{y}}{x ^{6}})

= ln (\frac{\frac{x}{y}}{x ^{6}}) - 4 ln (y)

Simplify

\frac{\frac{x}{y}}{x ^{6}} : \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})

4 ln (4 b)

ln (\frac{1}{2} \cdot e^{2 y})

ln (c x^{7})

ln (5) + ln (10)

1 0^{\frac{1}{2}} lo g_{10} (5)

What is ln(x)-ln(y)-6ln(x)-4ln(y) ?
The solution to ln(x)-ln(y)-6ln(x)-4ln(y) is -ln(x^5y^5)