Simplify/Condense log base 2 of 4x^2+8x+4

Problem

(log_2)(4*x^2+8*x+4)

Solution

1. Factor out the greatest common factor of 4 from the quadratic expression inside the logarithm.

(log_2)(4*(x^2+2*x+1))

2. Recognize that the expression x^2+2*x+1 is a perfect square trinomial, which factors into (x+1)^2

(log_2)(4*(x+1)^2)

3. Apply the product rule for logarithms, (log_b)(M*N)=(log_b)(M)+(log_b)(N) to separate the constant and the variable term.

(log_2)(4)+(log_2)((x+1)^2)

4. Evaluate the constant term (log_2)(4) by determining the power to which 2 must be raised to get 4

2+(log_2)((x+1)^2)

5. Apply the power rule for logarithms, (log_b)(M^p)=p*(log_b)(M) to move the exponent in front of the logarithm.

2+2*(log_2)(x+1)

Final Answer

(log_2)(4*x^2+8*x+4)=2+2*(log_2)(x+1)

Want more problems? Check here!