What is ln(x/(x-2))+ln((x+2)/x)-ln(x^2-4) ?

The solution to ln(x/(x-2))+ln((x+2)/x)-ln(x^2-4) is -ln((x-2)^2)

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simplify ln(x/(x-2))+ln((x+2)/x)-ln(x^2-4)

simplify

ln (\frac{x}{x - 2}) + ln (\frac{x + 2}{x}) - ln (x^{2} - 4)

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

Solution steps

ln (\frac{x}{x - 2}) + ln (\frac{x + 2}{x}) - ln (x^{2} - 4)

Apply log rule:

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

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

= ln (\frac{x}{x - 2} \cdot \frac{x + 2}{x}) - ln (x^{2} - 4)

Simplify

\frac{x}{x - 2} \cdot \frac{x + 2}{x} : \frac{x + 2}{x - 2}

= ln (\frac{x + 2}{x - 2}) - ln (x^{2} - 4)

Apply log rule:

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

ln (\frac{x + 2}{x - 2}) - ln (x^{2} - 4) = ln (\frac{\frac{x + 2}{x - 2}}{x ^{2} - 4})

= ln (\frac{\frac{x + 2}{x - 2}}{x ^{2} - 4})

Simplify

\frac{\frac{x + 2}{x - 2}}{x ^{2} - 4} : \frac{1}{( x - 2 ) ^{2}}

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

Apply log rule:

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

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

3 lo g_{10} (x) - 2 lo g_{10} (y)

lo g_{5} (\frac{8}{x})

\frac{1}{2} lo g_{7} (x) + lo g_{7} (y)

6 lo g_{b} (q) - lo g_{b} (r)

lo g_{10} (3) - lo g_{10} (2)

What is ln(x/(x-2))+ln((x+2)/x)-ln(x^2-4) ?
The solution to ln(x/(x-2))+ln((x+2)/x)-ln(x^2-4) is -ln((x-2)^2)