Simplify/Condense

Problem

ln(x/(x−4))+ln((x+4)/x)−ln(x2−16)

Solution

1. Apply the product rule for logarithms to the first two terms. The rule ln(a)+ln(b)=ln(a*b) allows us to combine the additions into a single logarithm of a product.

ln(x/(x−4)⋅(x+4)/x)−ln(x2−16)

2. Simplify the product inside the first logarithm. The x in the numerator and denominator cancel out.

ln((x+4)/(x−4))−ln(x2−16)

3. Apply the quotient rule for logarithms. The rule ln(a)−ln(b)=ln(a/b) allows us to combine the subtraction into a single logarithm of a quotient.

ln((x+4)/(x−4)/(x2−16))

4. Factor the denominator to simplify the complex fraction. Recognize that x2−16 is a difference of squares.

ln((x+4)/(x−4)/((x−4)*(x+4)))

5. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

ln((x+4)/(x−4)⋅1/((x−4)*(x+4)))

6. Cancel common terms in the numerator and denominator. The (x+4) terms cancel out.

ln(1/((x−4)2))

7. Apply power rules to rewrite the expression if desired. Using ln(1/a)=−ln(a) we can express the result in a more compact form.

−ln((x−4)2)

−2*ln(x−4)

Final Answer

ln(x/(x−4))+ln((x+4)/x)−ln(x2−16)=−2*ln(x−4)

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