What is 3log_{b}(z)+2(log_{b}(y)-5log_{b}(x)) ?

The solution to 3log_{b}(z)+2(log_{b}(y)-5log_{b}(x)) is log_{b}((z^3y^2)/(x^{10)})

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simplify 3log_{b}(z)+2(log_{b}(y)-5log_{b}(x))

simplify

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

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

Solution steps

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

Apply log rule:

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

5 lo g_{b} (x) = lo g_{b} (x^{5})

= 3 lo g_{b} (z) + 2 (lo g_{b} (y) - lo g_{b} (x^{5}))

Apply log rule:

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

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

= 3 lo g_{b} (z) + 2 lo g_{b} (\frac{y}{x ^{5}})

Apply log rule:

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

3 lo g_{b} (z) = lo g_{b} (z^{3})

= lo g_{b} (z^{3}) + 2 lo g_{b} (\frac{y}{x ^{5}})

Apply log rule:

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

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

= lo g_{b} (z^{3}) + lo g_{b} ((\frac{y}{x ^{5}})^{2})

Apply log rule:

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

lo g_{b} (z^{3}) + lo g_{b} ((\frac{y}{x ^{5}})^{2}) = lo g_{b} ((\frac{y}{x ^{5}})^{2} z^{3})

= lo g_{b} ((\frac{y}{x ^{5}})^{2} z^{3})

Simplify

z^{3} (\frac{y}{x ^{5}})^{2} : \frac{z ^{3} y ^{2}}{x ^{10}}

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

- \frac{( 8.314 \cdot 298 )}{96485} ln (\frac{0.002}{4})

lo g_{\frac{1}{2}} (\frac{1}{80})

4.74 + lo g_{10} (\frac{0.1}{0.175})

(75.3) ln (\frac{30}{100})

ln (\frac{x}{139})

What is 3log_{b}(z)+2(log_{b}(y)-5log_{b}(x)) ?
The solution to 3log_{b}(z)+2(log_{b}(y)-5log_{b}(x)) is log_{b}((z^3y^2)/(x^{10)})