What is 2log_{e}(x)-4log_{e}(1/y)-3log_{e}(xy) ?

The solution to 2log_{e}(x)-4log_{e}(1/y)-3log_{e}(xy) is ln(y/x)

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simplify 2log_{e}(x)-4log_{e}(1/y)-3log_{e}(xy)

simplify

2 lo g_{e} (x) - 4 lo g_{e} (\frac{1}{y}) - 3 lo g_{e} (x y)

ln (\frac{y}{x})

Solution steps

2 lo g_{e} (x) - 4 lo g_{e} (\frac{1}{y}) - 3 lo g_{e} (x y)

Apply log rule:

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

2 lo g_{e} (x) = lo g_{e} (x^{2})

= lo g_{e} (x^{2}) - 4 lo g_{e} (\frac{1}{y}) - 3 lo g_{e} (x y)

Apply log rule:

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

4 lo g_{e} (\frac{1}{y}) = lo g_{e} ((\frac{1}{y})^{4})

= lo g_{e} (x^{2}) - lo g_{e} ((\frac{1}{y})^{4}) - 3 lo g_{e} (x y)

Apply log rule:

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

lo g_{e} (x^{2}) - lo g_{e} ((\frac{1}{y})^{4}) = lo g_{e} \frac{x ^{2}}{( \frac{1}{y} ) ^{4}}

= lo g_{e} \frac{x ^{2}}{( \frac{1}{y} ) ^{4}} - 3 lo g_{e} (x y)

Simplify

\frac{x ^{2}}{( \frac{1}{y} ) ^{4}} : x^{2} y^{4}

= lo g_{e} (x^{2} y^{4}) - 3 lo g_{e} (x y)

Apply log rule:

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

3 lo g_{e} (x y) = lo g_{e} ((x y)^{3})

= lo g_{e} (x^{2} y^{4}) - lo g_{e} ((x y)^{3})

Apply log rule:

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

lo g_{e} (x^{2} y^{4}) - lo g_{e} ((x y)^{3}) = lo g_{e} (\frac{x ^{2} y ^{4}}{( x y ) ^{3}})

= lo g_{e} (\frac{x ^{2} y ^{4}}{( x y ) ^{3}})

Simplify

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

= lo g_{e} (\frac{y}{x})

Apply log rule:

lo g_{e} (x) = ln (x)

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

5 ln (x - 6) - ln (x^{2} - 10 x + 24) + 2 ln (x - 4)

ln (\frac{( \frac{17}{3} )}{19})

2 lo g_{3} (x) + lo g_{3} (a)

\frac{1}{6} ((ln ∣ 1 - 3 ∣ - ln ∣ 0 - 3 ∣) - (ln ∣ 1 + 3 ∣ - ln ∣ 0 + 3 ∣))

- lo g_{10} (6.38 \cdot 1 0^{- 6})

What is 2log_{e}(x)-4log_{e}(1/y)-3log_{e}(xy) ?
The solution to 2log_{e}(x)-4log_{e}(1/y)-3log_{e}(xy) is ln(y/x)