write the expression as a single logarithm express powers as factors `ln(x/{x-8}) + ln({x+8} /x) - ln(x^2-64)` must show work

Expert Answers

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To simplify this expression, we need to use the multiplication rule and division rule for logarithms:

`ln(x/{x-8})+ln({x+8}/x)-ln(x^2-64)`   combine + terms with multiplication

`=ln({x(x+8)}/{(x-8)x})-ln((x-8)(x+8))`   cancel common factors

`=ln({x+8}/{x-8})-ln((x-8)(x+8))`   combine - terms with division

`=ln({(x+8)}/{(x-8)(x-8)(x+8)})`    cancel common factors

`=ln(1/(x-8)^2)`    write as negative exponent

`=ln(x-8)^{-2}`   use power rule

`=-2ln(x-8)`

The expression simplifies to `-2ln(x-8)` .

 

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