What is 2log_{10}(x)-3log_{10}(xy)-4log_{10}(y) ?

The solution to 2log_{10}(x)-3log_{10}(xy)-4log_{10}(y) is -log_{10}(x)-7log_{10}(y)

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simplify 2log_{10}(x)-3log_{10}(xy)-4log_{10}(y)

simplify

2 lo g_{10} (x) - 3 lo g_{10} (x y) - 4 lo g_{10} (y)

- lo g_{10} (x) - 7 lo g_{10} (y)

Solution steps

2 lo g_{10} (x) - 3 lo g_{10} (x y) - 4 lo g_{10} (y)

2 lo g_{10} (x) = lo g_{10} (x^{2})

- 3 lo g_{10} (x y) = - lo g_{10} ((x y)^{3})

- 4 lo g_{10} (y) = - lo g_{10} (y^{4})

= lo g_{10} (x^{2}) - lo g_{10} ((x y)^{3}) - lo g_{10} (y^{4})

Apply log rule:

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

lo g_{10} (x^{2}) - lo g_{10} ((x y)^{3}) = lo g_{10} (\frac{x ^{2}}{( x y ) ^{3}})

= lo g_{10} (\frac{x ^{2}}{( x y ) ^{3}}) - lo g_{10} (y^{4})

Apply log rule:

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

lo g_{10} (\frac{x ^{2}}{( x y ) ^{3}}) - lo g_{10} (y^{4}) = lo g_{10} \frac{\frac{x ^{2}}{( x y ) ^{3}}}{y ^{4}}

= lo g_{10} \frac{\frac{x ^{2}}{( x y ) ^{3}}}{y ^{4}}

Simplify

\frac{\frac{x ^{2}}{( x y ) ^{3}}}{y ^{4}} : \frac{1}{x y ^{7}}

= lo g_{10} (\frac{1}{x y ^{7}})

Simplify

\frac{1}{x y ^{7}} : (x y^{7})^{- 1}

= lo g_{10} ((x y^{7})^{- 1})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), assuming x > 0

lo g_{10} ((x y^{7})^{- 1}) = - 1 \cdot lo g_{10} (x y^{7})

= - 1 \cdot lo g_{10} (x y^{7})

Apply rule:

1 \cdot a = a

= - lo g_{10} (x y^{7})

Apply log rule:

lo g_{a} (x y) = lo g_{a} (x) + lo g_{a} (y)

lo g_{10} (x y^{7}) = lo g_{10} (x) + lo g_{10} (y^{7})

= - (lo g_{10} (x) + lo g_{10} (y^{7}))

Simplify

lo g_{10} (y^{7}) : 7 lo g_{10} (y)

= - (lo g_{10} (x) + 7 lo g_{10} (y))

Apply the distributive law:

- (a + b) = - a - b

- (lo g_{10} (x) + 7 lo g_{10} (y)) = - lo g_{10} (x) - 7 lo g_{10} (y)

= - lo g_{10} (x) - 7 lo g_{10} (y)

lo g_{4} (x^{2} - 64) - 6 lo g_{4} (x + 8)

5 ln (x) + \frac{1}{2} ln (y) - 7 ln (z)

lo g_{10} (4) + lo g_{10} (3) - lo g_{10} (2)

ln (x^{6} y^{3})

- lo g_{10} (2 \cdot 1 0^{- 3})

What is 2log_{10}(x)-3log_{10}(xy)-4log_{10}(y) ?
The solution to 2log_{10}(x)-3log_{10}(xy)-4log_{10}(y) is -log_{10}(x)-7log_{10}(y)