What is ln(a)+2ln(a+b)-4ln(ab) ?

The solution to ln(a)+2ln(a+b)-4ln(ab) is ln(((a+b)^2)/(a^3b^4))

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simplify ln(a)+2ln(a+b)-4ln(ab)

simplify

ln (a) + 2 ln (a + b) - 4 ln (ab)

ln (\frac{( a + b ) ^{2}}{a ^{3} b ^{4}})

Solution steps

ln (a) + 2 ln (a + b) - 4 ln (ab)

Apply log rule:

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

2 ln (a + b) = ln ((a + b)^{2})

= ln (a) + ln ((a + b)^{2}) - 4 ln (ab)

Apply log rule:

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

ln (a) + ln ((a + b)^{2}) = ln (a (a + b)^{2})

= ln (a (a + b)^{2}) - 4 ln (ab)

Apply log rule:

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

4 ln (ab) = ln ((ab)^{4})

= ln (a (a + b)^{2}) - ln ((ab)^{4})

Apply log rule:

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

ln (a (a + b)^{2}) - ln ((ab)^{4}) = ln (\frac{a ( a + b ) ^{2}}{( ab ) ^{4}})

= ln (\frac{a ( a + b ) ^{2}}{( ab ) ^{4}})

Simplify

\frac{a ( a + b ) ^{2}}{( ab ) ^{4}} : \frac{( a + b ) ^{2}}{a ^{3} b ^{4}}

= ln (\frac{( a + b ) ^{2}}{a ^{3} b ^{4}})

lo g_{9} (4) + lo g_{9} (5)

lo g_{3} (y) + lo g_{3} (y + 4)

lo g_{9} (x y)

\frac{1}{3} (lo g_{4} (x) - lo g_{4} (y))

ln (x e^{- 2 x})

What is ln(a)+2ln(a+b)-4ln(ab) ?
The solution to ln(a)+2ln(a+b)-4ln(ab) is ln(((a+b)^2)/(a^3b^4))