Simplify/Condense log of x^2+3x+2-2 log of x+1

Problem

(log_)(x2+3*x+2)−2*(log_)(x+1)

Solution

1. Factor the quadratic expression inside the first logarithm.

x2+3*x+2=(x+1)*(x+2)

2. Apply the power property of logarithms, n*(log_)(a)=(log_)(an) to the second term.

2*(log_)(x+1)=(log_)((x+1)2)

3. Substitute these expressions back into the original problem.

(log_)((x+1)*(x+2))−(log_)((x+1)2)

4. Apply the quotient property of logarithms, (log_)(a)−(log_)(b)=(log_)(a/b)

(log_)(((x+1)*(x+2))/((x+1)2))

5. Simplify the fraction by canceling the common factor (x+1) from the numerator and denominator.

(log_)((x+2)/(x+1))

Final Answer

(log_)(x2+3*x+2)−2*(log_)(x+1)=(log_)((x+2)/(x+1))

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