What is ln(x/(x-6))+ln((x+6)/x)-ln(x^2-36) ?

The solution to ln(x/(x-6))+ln((x+6)/x)-ln(x^2-36) is -ln((x-6)^2)

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simplify ln(x/(x-6))+ln((x+6)/x)-ln(x^2-36)

simplify

ln (\frac{x}{x - 6}) + ln (\frac{x + 6}{x}) - ln (x^{2} - 36)

- ln ((x - 6)^{2})

Solution steps

ln (\frac{x}{x - 6}) + ln (\frac{x + 6}{x}) - ln (x^{2} - 36)

Apply log rule:

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

ln (\frac{x}{x - 6}) + ln (\frac{x + 6}{x}) = ln (\frac{x}{x - 6} \cdot \frac{x + 6}{x})

= ln (\frac{x}{x - 6} \cdot \frac{x + 6}{x}) - ln (x^{2} - 36)

Simplify

\frac{x}{x - 6} \cdot \frac{x + 6}{x} : \frac{x + 6}{x - 6}

= ln (\frac{x + 6}{x - 6}) - ln (x^{2} - 36)

Apply log rule:

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

ln (\frac{x + 6}{x - 6}) - ln (x^{2} - 36) = ln (\frac{\frac{x + 6}{x - 6}}{x ^{2} - 36})

= ln (\frac{\frac{x + 6}{x - 6}}{x ^{2} - 36})

Simplify

\frac{\frac{x + 6}{x - 6}}{x ^{2} - 36} : \frac{1}{( x - 6 ) ^{2}}

= ln (\frac{1}{( x - 6 ) ^{2}})

Apply log rule:

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

= - ln ((x - 6)^{2})

ln (\frac{x}{x + 1})

lo g_{10} (x) - \frac{1}{2} lo g_{10} (y) + 7 lo g_{10} (z)

3 ln (x) + 2 ln (y) - 4 ln (z)

lo g_{10} (\frac{x ^{3}}{2})

lo g_{11} (11 x)

What is ln(x/(x-6))+ln((x+6)/x)-ln(x^2-36) ?
The solution to ln(x/(x-6))+ln((x+6)/x)-ln(x^2-36) is -ln((x-6)^2)