What is 2log_{10}(x)-(3log_{10}(y)+2log_{10}(z)) ?

The solution to 2log_{10}(x)-(3log_{10}(y)+2log_{10}(z)) is log_{10}((x^2)/(y^3z^2))

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simplify 2log_{10}(x)-(3log_{10}(y)+2log_{10}(z))

simplify

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

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

Solution steps

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

Apply the distributive law:

- (a + b) = - a - b

- (3 lo g_{10} (y) + 2 lo g_{10} (z)) = - 3 lo g_{10} (y) - 2 lo g_{10} (z)

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

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

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

- 2 lo g_{10} (z) = - lo g_{10} (z^{2})

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{\frac{x ^{2}}{y ^{3}}}{z ^{2}} : \frac{x ^{2}}{y ^{3} z ^{2}}

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

ln (x) + 2 ln (y) - ln (z)

- 2 ln (3) + 2 ln (2)

ln (3 x) - ln (2 x)

ln (10) - ln (3)

What is 2log_{10}(x)-(3log_{10}(y)+2log_{10}(z)) ?
The solution to 2log_{10}(x)-(3log_{10}(y)+2log_{10}(z)) is log_{10}((x^2)/(y^3z^2))