Questions: Solve for (x) [ log 2(x+7)=2-log 2(x+4) ] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". [ x= ]

Solution Steps

To solve the equation \(\log_{2}(x+7) = 2 - \log_{2}(x+4)\), we can use properties of logarithms to combine and simplify the equation. Here are the steps:

1. Use the property of logarithms that states \(\log_{a}(b) - \log_{a}(c) = \log_{a}(\frac{b}{c})\).
2. Combine the logarithmic terms on one side of the equation.
3. Convert the logarithmic equation to an exponential equation.
4. Solve the resulting algebraic equation for \(x\).
5. Check the solutions to ensure they are valid within the domain of the original logarithmic functions.

Given the equation: \[ \log_{2}(x+7) = 2 - \log_{2}(x+4) \] we can use the property of logarithms that states \(\log_{a}(b) - \log_{a}(c) = \log_{a}\left(\frac{b}{c}\right)\) to combine the logarithmic terms. First, move \(\log_{2}(x+4)\) to the left side: \[ \log_{2}(x+7) + \log_{2}(x+4) = 2 \]

Using the property \(\log_{a}(b) + \log_{a}(c) = \log_{a}(bc)\), we combine the logarithms: \[ \log_{2}((x+7)(x+4)) = 2 \]

Convert the logarithmic equation to its exponential form: \[ (x+7)(x+4) = 2^2 \] \[ (x+7)(x+4) = 4 \]

Expand and simplify the quadratic equation: \[ x^2 + 11x + 28 = 4 \] \[ x^2 + 11x + 24 = 0 \] Solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\): \[ x = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \] \[ x = \frac{-11 \pm \sqrt{121 - 96}}{2} \] \[ x = \frac{-11 \pm \sqrt{25}}{2} \] \[ x = \frac{-11 \pm 5}{2} \] This gives us two solutions: \[ x = \frac{-11 + 5}{2} = -3 \] \[ x = \frac{-11 - 5}{2} = -8 \]

Check the solutions to ensure they are within the domain of the original logarithmic functions:

• For \(x = -3\): \[ \log_{2}(-3+7) = \log_{2}(4) \quad \text{and} \quad \log_{2}(-3+4) = \log_{2}(1) \] Both arguments are positive, so \(x = -3\) is valid.
• For \(x = -8\): \[ \log_{2}(-8+7) = \log_{2}(-1) \] The argument \(-1\) is not valid for a logarithm, so \(x = -8\) is not a valid solution.

Final Answer