What is ln(x/(x-4))+ln((x+4)/x)-ln(x^2-16) ?

The solution to ln(x/(x-4))+ln((x+4)/x)-ln(x^2-16) is -ln((x-4)^2)

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simplify ln(x/(x-4))+ln((x+4)/x)-ln(x^2-16)

simplify

ln (\frac{x}{x - 4}) + ln (\frac{x + 4}{x}) - ln (x^{2} - 16)

- ln ((x - 4)^{2})

Solution steps

ln (\frac{x}{x - 4}) + ln (\frac{x + 4}{x}) - ln (x^{2} - 16)

Apply log rule:

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

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

= ln (\frac{x}{x - 4} \cdot \frac{x + 4}{x}) - ln (x^{2} - 16)

Simplify

\frac{x}{x - 4} \cdot \frac{x + 4}{x} : \frac{x + 4}{x - 4}

= ln (\frac{x + 4}{x - 4}) - ln (x^{2} - 16)

Apply log rule:

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

ln (\frac{x + 4}{x - 4}) - ln (x^{2} - 16) = ln (\frac{\frac{x + 4}{x - 4}}{x ^{2} - 16})

= ln (\frac{\frac{x + 4}{x - 4}}{x ^{2} - 16})

Simplify

\frac{\frac{x + 4}{x - 4}}{x ^{2} - 16} : \frac{1}{( x - 4 ) ^{2}}

= ln (\frac{1}{( x - 4 ) ^{2}})

Apply log rule:

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

= - ln ((x - 4)^{2})

3 lo g_{10} (x) + lo g_{10} (2)

ln (4 x^{2})

- lo g_{10} (x + 6) + lo g_{10} (x - 5)

lo g_{10} (\frac{2 x}{5})

lo g_{10} (x) + lo g_{10} (y) - lo g_{10} (z)

What is ln(x/(x-4))+ln((x+4)/x)-ln(x^2-16) ?
The solution to ln(x/(x-4))+ln((x+4)/x)-ln(x^2-16) is -ln((x-4)^2)