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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Simplify

\frac{x}{x - 3} \cdot \frac{x + 3}{x} : \frac{x + 3}{x - 3}

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

Apply log rule:

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

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

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

Simplify

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

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

Apply log rule:

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

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

lo g_{10} (\frac{3}{x})

lo g_{10} (6) - lo g_{10} (3)

lo g_{10} (2 ab)

lo g_{10} (a b^{2})

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