What is 9ln(x)-2ln(y)+5ln(z^3) ?

The solution to 9ln(x)-2ln(y)+5ln(z^3) is ln((x^9z^{15})/(y^2))

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simplify 9ln(x)-2ln(y)+5ln(z^3)

simplify

9 ln (x) - 2 ln (y) + 5 ln (z^{3})

ln (\frac{x ^{9} z ^{15}}{y ^{2}})

Solution steps

9 ln (x) - 2 ln (y) + 5 ln (z^{3})

Apply log rule:

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

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

= ln (x^{9}) - 2 ln (y) + 5 ln (z^{3})

Apply log rule:

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

2 ln (y) = ln (y^{2})

= ln (x^{9}) - ln (y^{2}) + 5 ln (z^{3})

Apply log rule:

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

ln (x^{9}) - ln (y^{2}) = ln (\frac{x ^{9}}{y ^{2}})

= ln (\frac{x ^{9}}{y ^{2}}) + 5 ln (z^{3})

Apply log rule:

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

5 ln (z^{3}) = ln ((z^{3})^{5})

= ln (\frac{x ^{9}}{y ^{2}}) + ln ((z^{3})^{5})

Apply log rule:

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

ln (\frac{x ^{9}}{y ^{2}}) + ln ((z^{3})^{5}) = ln (\frac{x ^{9}}{y ^{2}} (z^{3})^{5})

= ln (\frac{x ^{9}}{y ^{2}} (z^{3})^{5})

Simplify

\frac{x ^{9}}{y ^{2}} (z^{3})^{5} : \frac{x ^{9} z ^{15}}{y ^{2}}

= ln (\frac{x ^{9} z ^{15}}{y ^{2}})

lo g_{4} (\frac{1}{16} x^{2} y^{3})

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lo g_{7} (9) + lo g_{7} (8)

f (x) = lo g_{10} (2 x) - lo g_{10} (8 x^{3})

ln (x y z^{6})

What is 9ln(x)-2ln(y)+5ln(z^3) ?
The solution to 9ln(x)-2ln(y)+5ln(z^3) is ln((x^9z^{15})/(y^2))