# 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

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Expert Answers

lfryerda | Certified Educator

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)` .**