What is simplify-5ln(x)-9ln(y)+1/9 ln(z) ?

The solution to simplify-5ln(x)-9ln(y)+1/9 ln(z) is ln((z^{1/9})/(x^5y^9))

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simplify-5ln(x)-9ln(y)+1/9 ln(z)

simplify

- 5 ln (x) - 9 ln (y) + \frac{1}{9} ln (z)

ln (\frac{z ^{\frac{1}{9}}}{x ^{5} y ^{9}})

Solution steps

- 5 ln (x) - 9 ln (y) + \frac{1}{9} ln (z)

Apply log rule:

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

5 ln (x) = ln (x^{5})

= - ln (x^{5}) - 9 ln (y) + \frac{1}{9} ln (z)

Apply log rule:

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

9 ln (y) = ln (y^{9})

= - ln (x^{5}) - ln (y^{9}) + \frac{1}{9} ln (z)

Rewrite

- ln (x^{5}) - ln (y^{9})

- (ln (x^{5}) + ln (y^{9}))

= - (ln (x^{5}) + ln (y^{9})) + \frac{1}{9} ln (z)

Apply log rule:

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

ln (x^{5}) + ln (y^{9}) = ln (x^{5} y^{9})

= - ln (x^{5} y^{9}) + \frac{1}{9} ln (z)

Apply log rule:

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

\frac{1}{9} ln (z) = ln (z^{\frac{1}{9}})

= - ln (x^{5} y^{9}) + ln (z^{\frac{1}{9}})

Apply log rule:

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

ln (z^{\frac{1}{9}}) - ln (x^{5} y^{9}) = ln (\frac{z ^{\frac{1}{9}}}{x ^{5} y ^{9}})

= ln (\frac{z ^{\frac{1}{9}}}{x ^{5} y ^{9}})

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

- ln \frac{8}{3} + \frac{8}{3} (0) + ln \frac{2}{3} + \frac{8}{3} (0)

1 + 3 ln (x) - 4 ln (y) + \frac{1}{2} ln (z)

3 lo g_{10} (x) + lo g_{10} (2) - lo g_{10} (y)

lo g_{4} (\frac{f}{64})

What is simplify-5ln(x)-9ln(y)+1/9 ln(z) ?
The solution to simplify-5ln(x)-9ln(y)+1/9 ln(z) is ln((z^{1/9})/(x^5y^9))