Questions: Find the value of (x) which satisfies the following equation. [ log 2(x-1)+log 2(x+5)=4 ]

Solution Steps

To solve the given logarithmic equation, we can use the properties of logarithms to combine the two logarithmic terms into one. Specifically, we use the property that \(\log_b(a) + \log_b(c) = \log_b(ac)\). After combining the logs, we can then exponentiate both sides to solve for \(x\).

1. Combine the logarithmic terms using the property of logarithms.
2. Exponentiate both sides to remove the logarithm.
3. Solve the resulting quadratic equation for \(x\).

Given the equation: \[ \log_{2}(x-1) + \log_{2}(x+5) = 4 \] we use the property of logarithms \(\log_b(a) + \log_b(c) = \log_b(ac)\) to combine the terms: \[ \log_{2}((x-1)(x+5)) = 4 \]

To remove the logarithm, we exponentiate both sides with base 2: \[ (x-1)(x+5) = 2^4 \] \[ (x-1)(x+5) = 16 \]

Expand the left-hand side and set the equation to zero: \[ x^2 + 5x - x - 5 = 16 \] \[ x^2 + 4x - 5 = 16 \] \[ x^2 + 4x - 21 = 0 \]

Solve the quadratic equation \(x^2 + 4x - 21 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\): \[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-21)}}{2 \cdot 1} \] \[ x = \frac{-4 \pm \sqrt{16 + 84}}{2} \] \[ x = \frac{-4 \pm \sqrt{100}}{2} \] \[ x = \frac{-4 \pm 10}{2} \] \[ x = 3 \quad \text{or} \quad x = -7 \]

Check the solutions in the original equation:

• For \(x = 3\): \[ \log_{2}(3-1) + \log_{2}(3+5) = \log_{2}(2) + \log_{2}(8) = 1 + 3 = 4 \] This is valid.
• For \(x = -7\): \[ \log_{2}(-7-1) + \log_{2}(-7+5) \] This is not valid because the logarithm of a negative number is undefined.

Final Answer