What is 6log_{6}(z)-1/4 log_{6}(y)+7log_{6}(x) ?

The solution to 6log_{6}(z)-1/4 log_{6}(y)+7log_{6}(x) is log_{6}((z^6x^7y^{3/4})/y)

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simplify 6log_{6}(z)-1/4 log_{6}(y)+7log_{6}(x)

simplify

6 lo g_{6} (z) - \frac{1}{4} lo g_{6} (y) + 7 lo g_{6} (x)

lo g_{6} (\frac{z ^{6} x ^{7} y ^{\frac{3}{4}}}{y})

Solution steps

6 lo g_{6} (z) - \frac{1}{4} lo g_{6} (y) + 7 lo g_{6} (x)

Apply log rule:

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

6 lo g_{6} (z) = lo g_{6} (z^{6})

= lo g_{6} (z^{6}) - \frac{1}{4} lo g_{6} (y) + 7 lo g_{6} (x)

Apply log rule:

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

\frac{1}{4} lo g_{6} (y) = lo g_{6} (y^{\frac{1}{4}})

= lo g_{6} (z^{6}) - lo g_{6} (y^{\frac{1}{4}}) + 7 lo g_{6} (x)

Apply log rule:

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

lo g_{6} (z^{6}) - lo g_{6} (y^{\frac{1}{4}}) = lo g_{6} (\frac{z ^{6}}{y ^{\frac{1}{4}}})

= lo g_{6} (\frac{z ^{6}}{y ^{\frac{1}{4}}}) + 7 lo g_{6} (x)

Simplify

\frac{z ^{6}}{y ^{\frac{1}{4}}} : \frac{z ^{6} y ^{\frac{3}{4}}}{y}

= lo g_{6} (\frac{z ^{6} y ^{\frac{3}{4}}}{y}) + 7 lo g_{6} (x)

Apply log rule:

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

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

= lo g_{6} (\frac{z ^{6} y ^{\frac{3}{4}}}{y}) + lo g_{6} (x^{7})

Apply log rule:

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

lo g_{6} (\frac{z ^{6} y ^{\frac{3}{4}}}{y}) + lo g_{6} (x^{7}) = lo g_{6} (\frac{z ^{6} y ^{\frac{3}{4}}}{y} x^{7})

= lo g_{6} (\frac{z ^{6} y ^{\frac{3}{4}}}{y} x^{7})

Simplify

\frac{z ^{6} y ^{\frac{3}{4}}}{y} x^{7} : \frac{z ^{6} x ^{7} y ^{\frac{3}{4}}}{y}

= lo g_{6} (\frac{z ^{6} x ^{7} y ^{\frac{3}{4}}}{y})

\frac{1}{3} (ln (9) - ln (2))

2 x ln (3 x) - 4 x - 4 ln (3 x) + 8

ln (\frac{x}{1 e ^{6}})

- 160 ln (x) - 160 ln (y)

23 x + ln (23 x)

What is 6log_{6}(z)-1/4 log_{6}(y)+7log_{6}(x) ?
The solution to 6log_{6}(z)-1/4 log_{6}(y)+7log_{6}(x) is log_{6}((z^6x^7y^{3/4})/y)