Part 2: Algebraic Logarithm (5 points) Given the equation \( \log (2 x-3)+\log 8=\log (9 x+11) \), show all steps used to solve for \( x \).

To solve the equation \( \log (2x - 3) + \log 8 = \log (9x + 11) \), we can start by using the property of logarithms that states \( \log a + \log b = \log (ab) \). 1. Combine the logarithms on the left side: \[ \log (2x - 3) + \log 8 = \log (8(2x - 3)). \] So the equation becomes: \[ \log (8(2x - 3)) = \log (9x + 11). \] 2. Since the logarithms are equal, we can set the arguments equal to each other: \[ 8(2x - 3) = 9x + 11. \] 3. Distribute the 8 on the left side: \[ 16x - 24 = 9x + 11. \] 4. Rearranging the equation, we move \( 9x \) to the left side and \( -24 \) to the right side: \[ 16x - 9x = 11 + 24. \] 5. This simplifies to: \[ 7x = 35. \] 6. Divide both sides by 7 to solve for \( x \): \[ x = 5. \] 7. Lastly, we need to check if this value satisfies the original equation by substituting \( x = 5 \) back in: - Calculate \( 2(5) - 3 = 10 - 3 = 7 \). - Calculate \( 9(5) + 11 = 45 + 11 = 56 \). - Check the logarithmic terms: \[ \log(7) + \log(8) = \log(56) \] which is true since \( \log(7 \times 8) = \log(56) \). Therefore, the solution to the equation is \( x = 5 \).