What is 6ln(z)-1/7 ln(x)+6ln(y) ?

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

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simplify 6ln(z)-1/7 ln(x)+6ln(y)

simplify

6 ln (z) - \frac{1}{7} ln (x) + 6 ln (y)

ln (\frac{z ^{6} y ^{6} x ^{\frac{6}{7}}}{x})

Solution steps

6 ln (z) - \frac{1}{7} ln (x) + 6 ln (y)

Apply log rule:

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

6 ln (z) = ln (z^{6})

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

Apply log rule:

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

\frac{1}{7} ln (x) = ln (x^{\frac{1}{7}})

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

Apply log rule:

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

ln (z^{6}) - ln (x^{\frac{1}{7}}) = ln (\frac{z ^{6}}{x ^{\frac{1}{7}}})

= ln (\frac{z ^{6}}{x ^{\frac{1}{7}}}) + 6 ln (y)

Simplify

\frac{z ^{6}}{x ^{\frac{1}{7}}} : \frac{z ^{6} x ^{\frac{6}{7}}}{x}

= ln (\frac{z ^{6} x ^{\frac{6}{7}}}{x}) + 6 ln (y)

Apply log rule:

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

6 ln (y) = ln (y^{6})

= ln (\frac{z ^{6} x ^{\frac{6}{7}}}{x}) + ln (y^{6})

Apply log rule:

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

ln (\frac{z ^{6} x ^{\frac{6}{7}}}{x}) + ln (y^{6}) = ln (\frac{z ^{6} x ^{\frac{6}{7}}}{x} y^{6})

= ln (\frac{z ^{6} x ^{\frac{6}{7}}}{x} y^{6})

Simplify

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

= ln (\frac{z ^{6} y ^{6} x ^{\frac{6}{7}}}{x})

lo g_{7} (2 y)

lo g_{10} (\frac{11 x}{13})

lo g_{10} (1 + 3) \cdot lo g_{10} (\frac{4}{3})

lo g_{2} (\frac{8 x ^{2}}{y})

lo g_{3} (t) + lo g_{3} (s)

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