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

The solution to ln(x^2-9)-ln(2x^2-5x-3)-2ln(x+3) is -ln((2x+1)(x+3))

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

simplify

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

- ln ((2 x + 1) (x + 3))

Solution steps

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

Apply log rule:

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

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

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

Simplify

\frac{x ^{2} - 9}{2 x ^{2} - 5 x - 3} : \frac{x + 3}{2 x + 1}

= ln (\frac{x + 3}{2 x + 1}) - 2 ln (x + 3)

Apply log rule:

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

2 ln (x + 3) = ln ((x + 3)^{2})

= ln (\frac{x + 3}{2 x + 1}) - ln ((x + 3)^{2})

Apply log rule:

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

ln (\frac{x + 3}{2 x + 1}) - ln ((x + 3)^{2}) = ln (\frac{\frac{x + 3}{2 x + 1}}{( x + 3 ) ^{2}})

= ln (\frac{\frac{x + 3}{2 x + 1}}{( x + 3 ) ^{2}})

Simplify

\frac{\frac{x + 3}{2 x + 1}}{( x + 3 ) ^{2}} : \frac{1}{( 2 x + 1 ) ( x + 3 )}

= ln (\frac{1}{( 2 x + 1 ) ( x + 3 )})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln ((2 x + 1) (x + 3))

- lo g_{10} (2 \cdot 1 0^{- 14})

\frac{1}{2} ln (\frac{3}{2})

ln (4) + 2 ln (3)

ln (- 5 e)

ln (\frac{x ^{x}}{x ^{2} - 1})

What is ln(x^2-9)-ln(2x^2-5x-3)-2ln(x+3) ?
The solution to ln(x^2-9)-ln(2x^2-5x-3)-2ln(x+3) is -ln((2x+1)(x+3))