What is 12ln(x)+5ln(y)-11ln(z) ?

The solution to 12ln(x)+5ln(y)-11ln(z) is ln((x^{12}y^5)/(z^{11)})

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simplify 12ln(x)+5ln(y)-11ln(z)

simplify

12 ln (x) + 5 ln (y) - 11 ln (z)

ln (\frac{x ^{12} y ^{5}}{z ^{11}})

Solution steps

12 ln (x) + 5 ln (y) - 11 ln (z)

Apply log rule:

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

12 ln (x) = ln (x^{12})

= ln (x^{12}) + 5 ln (y) - 11 ln (z)

Apply log rule:

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

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

= ln (x^{12}) + ln (y^{5}) - 11 ln (z)

Apply log rule:

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

ln (x^{12}) + ln (y^{5}) = ln (x^{12} y^{5})

= ln (x^{12} y^{5}) - 11 ln (z)

Apply log rule:

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

11 ln (z) = ln (z^{11})

= ln (x^{12} y^{5}) - ln (z^{11})

Apply log rule:

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

ln (x^{12} y^{5}) - ln (z^{11}) = ln (\frac{x ^{12} y ^{5}}{z ^{11}})

= ln (\frac{x ^{12} y ^{5}}{z ^{11}})

ln (\frac{x + 4}{x ^{2} - 5 x + 6})

ln (x + y) + ln (x - y) - 2 ln (a)

ln ((e^{z} x^{2} y)^{3})

(ln (3 e))^{2} - 3 ln (3 e)

ln (9) + 4 ln (4)

What is 12ln(x)+5ln(y)-11ln(z) ?
The solution to 12ln(x)+5ln(y)-11ln(z) is ln((x^{12}y^5)/(z^{11)})