What is 2(log_{m}(x)-5log_{m}(y))+2log_{m}(z) ?

The solution to 2(log_{m}(x)-5log_{m}(y))+2log_{m}(z) is log_{m}((x^2z^2)/(y^{10)})

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simplify 2(log_{m}(x)-5log_{m}(y))+2log_{m}(z)

simplify

2 (lo g_{m} (x) - 5 lo g_{m} (y)) + 2 lo g_{m} (z)

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

Solution steps

2 (lo g_{m} (x) - 5 lo g_{m} (y)) + 2 lo g_{m} (z)

Apply log rule:

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

5 lo g_{m} (y) = lo g_{m} (y^{5})

= 2 (lo g_{m} (x) - lo g_{m} (y^{5})) + 2 lo g_{m} (z)

Apply log rule:

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

lo g_{m} (x) - lo g_{m} (y^{5}) = lo g_{m} (\frac{x}{y ^{5}})

= 2 lo g_{m} (\frac{x}{y ^{5}}) + 2 lo g_{m} (z)

Apply log rule:

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

2 lo g_{m} (\frac{x}{y ^{5}}) = lo g_{m} ((\frac{x}{y ^{5}})^{2})

= lo g_{m} ((\frac{x}{y ^{5}})^{2}) + 2 lo g_{m} (z)

Apply log rule:

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

2 lo g_{m} (z) = lo g_{m} (z^{2})

= lo g_{m} ((\frac{x}{y ^{5}})^{2}) + lo g_{m} (z^{2})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{m} ((\frac{x}{y ^{5}})^{2}) + lo g_{m} (z^{2}) = lo g_{m} ((\frac{x}{y ^{5}})^{2} z^{2})

= lo g_{m} ((\frac{x}{y ^{5}})^{2} z^{2})

Simplify

(\frac{x}{y ^{5}})^{2} z^{2} : \frac{x ^{2} z ^{2}}{y ^{10}}

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

lo g_{10} (0.98 \cdot 111.2 \cdot 1 0^{- 6} \cdot 2 π \cdot 200.27)

- \frac{8.5}{30} ln (\frac{1}{2})

3 lo g_{10} (6 - x) - lo g_{10} (72 - x^{3})

lo g_{10} (\frac{23.16}{2})

0.026 \cdot ln (\frac{0.4644}{8 \cdot 1 0 ^{- 16}})

What is 2(log_{m}(x)-5log_{m}(y))+2log_{m}(z) ?
The solution to 2(log_{m}(x)-5log_{m}(y))+2log_{m}(z) is log_{m}((x^2z^2)/(y^{10)})