What is ln(x^a2y^b(1-x-2y)^c) ?

The solution to ln(x^a2y^b(1-x-2y)^c) is aln(x)+ln(2)+bln(y)+cln(1-x-2y)

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simplify ln(x^a2y^b(1-x-2y)^c)

simplify

ln (x^{a} 2 y^{b} (1 - x - 2 y)^{c})

a ln (x) + ln (2) + b ln (y) + c ln (1 - x - 2 y)

Solution steps

ln (x^{a} \cdot 2 y^{b} (1 - x - 2 y)^{c})

Apply log rule:

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

ln (x^{a} \cdot 2 y^{b} (1 - x - 2 y)^{c}) = ln (x^{a}) + ln (2) + ln (y^{b}) + ln ((1 - x - 2 y)^{c})

= ln (x^{a}) + ln (2) + ln (y^{b}) + ln ((1 - x - 2 y)^{c})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= a ln (x) + ln (2) + ln (y^{b}) + ln ((1 - x - 2 y)^{c})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= a ln (x) + ln (2) + b ln (y) + ln ((1 - x - 2 y)^{c})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((1 - x - 2 y)^{c}) = c ln (1 - x - 2 y)

= a ln (x) + ln (2) + b ln (y) + c ln (1 - x - 2 y)

ln (\frac{1}{3} e^{3 x})

ln (\frac{9}{16})

ln (\frac{10 + y ^{4}}{4})

lo g_{10} (\frac{x ^{2} y ^{3}}{z ^{4}})

\frac{1}{16} ln (\frac{1}{4})

What is ln(x^a2y^b(1-x-2y)^c) ?
The solution to ln(x^a2y^b(1-x-2y)^c) is aln(x)+ln(2)+bln(y)+cln(1-x-2y)