Simplify/Condense

Problem

3*(log_a)(2*x+1)−2*(log_a)(2*x−1)+2

Solution

1. Apply the power property of logarithms, which states n*(log_b)(m)=(log_b)(mn) to the first two terms.

(log_a)((2*x+1)3)−(log_a)((2*x−1)2)+2

2. Rewrite the constant 2 as a logarithm with base a using the identity n=(log_a)(an)

(log_a)((2*x+1)3)−(log_a)((2*x−1)2)+(log_a)(a2)

3. Apply the quotient property of logarithms, (log_b)(m)−(log_b)(n)=(log_b)(m/n) to the first two terms.

(log_a)(((2*x+1)3)/((2*x−1)2))+(log_a)(a2)

4. Apply the product property of logarithms, (log_b)(m)+(log_b)(n)=(log_b)(m*n) to combine the remaining terms.

(log_a)((a2*(2*x+1)3)/((2*x−1)2))

Final Answer

3*(log_a)(2*x+1)−2*(log_a)(2*x−1)+2=(log_a)((a2*(2*x+1)3)/((2*x−1)2))

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