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

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

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

simplify

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

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

Solution steps

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

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

Simplify

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

= 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} (ab)

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

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

Simplify

\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))

ln (24) - 3 ln (2)

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

\frac{3}{8} \cdot ln (\frac{8}{3}) + \frac{5}{16} \cdot ln (\frac{16}{5}) + \frac{5}{16} \cdot ln (\frac{16}{5})

ln (\frac{y}{x} - 1) - ln (x)

- lo g_{10} (5.1 \cdot 1 0^{- 11})

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