What is ln(x/(x-5))+ln((x+5)/x)-ln(x^2-25) ?

The solution to ln(x/(x-5))+ln((x+5)/x)-ln(x^2-25) is -ln((x-5)^2)

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simplify ln(x/(x-5))+ln((x+5)/x)-ln(x^2-25)

simplify

ln (\frac{x}{x - 5}) + ln (\frac{x + 5}{x}) - ln (x^{2} - 25)

- ln ((x - 5)^{2})

Solution steps

ln (\frac{x}{x - 5}) + ln (\frac{x + 5}{x}) - ln (x^{2} - 25)

Apply log rule:

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

ln (\frac{x}{x - 5}) + ln (\frac{x + 5}{x}) = ln (\frac{x}{x - 5} \cdot \frac{x + 5}{x})

= ln (\frac{x}{x - 5} \cdot \frac{x + 5}{x}) - ln (x^{2} - 25)

Simplify

\frac{x}{x - 5} \cdot \frac{x + 5}{x} : \frac{x + 5}{x - 5}

= ln (\frac{x + 5}{x - 5}) - ln (x^{2} - 25)

Apply log rule:

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

ln (\frac{x + 5}{x - 5}) - ln (x^{2} - 25) = ln (\frac{\frac{x + 5}{x - 5}}{x ^{2} - 25})

= ln (\frac{\frac{x + 5}{x - 5}}{x ^{2} - 25})

Simplify

\frac{\frac{x + 5}{x - 5}}{x ^{2} - 25} : \frac{1}{( x - 5 ) ^{2}}

= ln (\frac{1}{( x - 5 ) ^{2}})

Apply log rule:

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

= - ln ((x - 5)^{2})

lo g_{b} (\frac{x}{yz})

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

\frac{2}{5} ln (\frac{15}{2})

lo g_{10} (\frac{z ^{5}}{x ^{3} y})

\frac{1}{2} lo g_{10} (x) + 3 lo g_{10} (x + 1)

What is ln(x/(x-5))+ln((x+5)/x)-ln(x^2-25) ?
The solution to ln(x/(x-5))+ln((x+5)/x)-ln(x^2-25) is -ln((x-5)^2)