What is 3log_{a}(7x^6)-1/8 log_{a}(5x+19) ?

The solution to 3log_{a}(7x^6)-1/8 log_{a}(5x+19) is log_{a}((343x^{18}(5x+19)^{7/8})/(5x+19))

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simplify 3log_{a}(7x^6)-1/8 log_{a}(5x+19)

simplify

3 lo g_{a} (7 x^{6}) - \frac{1}{8} lo g_{a} (5 x + 19)

lo g_{a} (\frac{343 x ^{18} ( 5 x + 19 ) ^{\frac{7}{8}}}{5 x + 19})

Solution steps

3 lo g_{a} (7 x^{6}) - \frac{1}{8} lo g_{a} (5 x + 19)

Apply log rule:

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

3 lo g_{a} (7 x^{6}) = lo g_{a} ((7 x^{6})^{3})

= lo g_{a} ((7 x^{6})^{3}) - \frac{1}{8} lo g_{a} (5 x + 19)

Apply log rule:

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

\frac{1}{8} lo g_{a} (5 x + 19) = lo g_{a} ((5 x + 19)^{\frac{1}{8}})

= lo g_{a} ((7 x^{6})^{3}) - lo g_{a} ((5 x + 19)^{\frac{1}{8}})

Apply log rule:

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

lo g_{a} ((7 x^{6})^{3}) - lo g_{a} ((5 x + 19)^{\frac{1}{8}}) = lo g_{a} (\frac{( 7 x ^{6} ) ^{3}}{( 5 x + 19 ) ^{\frac{1}{8}}})

= lo g_{a} (\frac{( 7 x ^{6} ) ^{3}}{( 5 x + 19 ) ^{\frac{1}{8}}})

Simplify

\frac{( 7 x ^{6} ) ^{3}}{( 5 x + 19 ) ^{\frac{1}{8}}} : \frac{343 x ^{18} ( 5 x + 19 ) ^{\frac{7}{8}}}{5 x + 19}

= lo g_{a} (\frac{343 x ^{18} ( 5 x + 19 ) ^{\frac{7}{8}}}{5 x + 19})

lo g_{\frac{a}{b}} (\frac{b}{a})

ln (\frac{8}{13})

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

lo g_{10} (\frac{z ^{4}}{5})

2 lo g_{3} (x) + 4 lo g_{3} (x - 1) + lo g_{3} (y)

What is 3log_{a}(7x^6)-1/8 log_{a}(5x+19) ?
The solution to 3log_{a}(7x^6)-1/8 log_{a}(5x+19) is log_{a}((343x^{18}(5x+19)^{7/8})/(5x+19))