Answer: x = 6 or x = 5

<p>x = 1/2 (5 + sqrt(25 + 4 e^2))</p>

<p><br></p><p>To solve the equation log(x) + log(x - 5) = 2, we can use logarithmic properties to simplify and solve for x.</p><p>Using the logarithmic property log(a) + log(b) = log(ab), we can rewrite the equation as:</p><p>log(x(x - 5)) = 2</p><p>Now, we can rewrite the equation in exponential form:</p><p>x(x - 5) = 10^2</p><p>Simplifying the right side:</p><p>x(x - 5) = 100</p><p>Expanding the left side:</p><p>x^2 - 5x = 100</p><p>Rearranging the equation to bring all terms to one side:</p><p>x^2 - 5x - 100 = 0</p><p>Now we have a quadratic equation. To solve for x, we can factor the equation or use the quadratic formula.</p><p>Factoring:</p><p>(x - 10)(x + 10) = 0</p><p>Setting each factor equal to zero:</p><p>x - 10 = 0 or x + 10 = 0</p><p>Solving for x:</p><p>x = 10 or x = -10</p><p>However, we need to check if the solutions satisfy the original equation.</p><p>When x = 10:</p><p>log(10) + log(10 - 5) = log(10) + log(5) = 1 + 0.69897 ≈ 1.69897</p><p>When x = -10:</p><p>The logarithm is undefined for negative values, so x = -10 is not a valid solution.</p><p>Therefore, the solution to the equation is x = 10.</p>