What is ln(x/(x-9))+ln((x+9)/x)-ln(x^2-81) ?

The solution to ln(x/(x-9))+ln((x+9)/x)-ln(x^2-81) is -ln((x-9)^2)

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simplify ln(x/(x-9))+ln((x+9)/x)-ln(x^2-81)

simplify

ln (\frac{x}{x - 9}) + ln (\frac{x + 9}{x}) - ln (x^{2} - 81)

- ln ((x - 9)^{2})

Solution steps

ln (\frac{x}{x - 9}) + ln (\frac{x + 9}{x}) - ln (x^{2} - 81)

Apply log rule:

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

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

= ln (\frac{x}{x - 9} \cdot \frac{x + 9}{x}) - ln (x^{2} - 81)

Simplify

\frac{x}{x - 9} \cdot \frac{x + 9}{x} : \frac{x + 9}{x - 9}

= ln (\frac{x + 9}{x - 9}) - ln (x^{2} - 81)

Apply log rule:

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

ln (\frac{x + 9}{x - 9}) - ln (x^{2} - 81) = ln (\frac{\frac{x + 9}{x - 9}}{x ^{2} - 81})

= ln (\frac{\frac{x + 9}{x - 9}}{x ^{2} - 81})

Simplify

\frac{\frac{x + 9}{x - 9}}{x ^{2} - 81} : \frac{1}{( x - 9 ) ^{2}}

= ln (\frac{1}{( x - 9 ) ^{2}})

Apply log rule:

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

= - ln ((x - 9)^{2})

x^{2} + 7 x + 6 > 0

16 y^{2} - 9

4 x^{2} + 9 x = - 3

\frac{k ( 4 - k )}{4} > 0

\frac{7 i}{4 + 4 i}

What is ln(x/(x-9))+ln((x+9)/x)-ln(x^2-81) ?
The solution to ln(x/(x-9))+ln((x+9)/x)-ln(x^2-81) is -ln((x-9)^2)