Questions: Solve for (x) : [ log 3 x+log 3(x+5)=5 x= ]

Solve for (x) : [ log 3 x+log 3(x+5)=5 x= ]
Transcript text: Solve for $x$ : \[ \begin{array}{l} \log _{3} x+\log _{3}(x+5)=5 \\ x= \end{array} \]
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Solution

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Solution Steps

To solve the equation \(\log_{3} x + \log_{3} (x+5) = 5\), we can use the properties of logarithms. Specifically, we can use the product rule for logarithms, which states that \(\log_{b} a + \log_{b} c = \log_{b} (a \cdot c)\). Applying this rule, we can combine the logarithms on the left side of the equation. Then, we can rewrite the equation in exponential form to solve for \(x\).

Step 1: Combine Logarithms

We start with the equation: \[ \log_{3} x + \log_{3} (x + 5) = 5 \] Using the product rule for logarithms, we can combine the left side: \[ \log_{3} (x(x + 5)) = 5 \]

Step 2: Convert to Exponential Form

Next, we convert the logarithmic equation to its exponential form: \[ x(x + 5) = 3^5 \] Calculating \(3^5\): \[ 3^5 = 243 \] Thus, we have: \[ x^2 + 5x - 243 = 0 \]

Step 3: Solve the Quadratic Equation

We can solve the quadratic equation \(x^2 + 5x - 243 = 0\) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 5\), and \(c = -243\). Plugging in these values: \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-243)}}{2 \cdot 1} \] Calculating the discriminant: \[ 5^2 - 4 \cdot 1 \cdot (-243) = 25 + 972 = 997 \] Thus, the solutions for \(x\) are: \[ x = \frac{-5 \pm \sqrt{997}}{2} \]

Final Answer

The solutions are: \[ x = \frac{-5 + \sqrt{997}}{2} \quad \text{and} \quad x = \frac{-5 - \sqrt{997}}{2} \] Since \(x\) must be positive, we take the positive solution: \[ \boxed{x = \frac{-5 + \sqrt{997}}{2}} \]

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