folve the equation \( 1+\log (2 x-9)=\log (3 x+5)-\log 2 \). 27. COMMERCIAI ARITHMETIC (II)

Solve the equation \( 1+\log (2x-9)=\log (3x+5)-\log 2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(1+\log_{10}{\left(2x-9\right)}=\log_{10}{\left(3x+5\right)}-\log_{10}{\left(2\right)}\) - step1: Find the domain: \(1+\log_{10}{\left(2x-9\right)}=\log_{10}{\left(3x+5\right)}-\log_{10}{\left(2\right)},x>\frac{9}{2}\) - step2: Transform the expression: \(1+\log_{10}{\left(2x-9\right)}=\log_{10}{\left(\frac{3x+5}{2}\right)}\) - step3: Rewrite in terms of common logarithms: \(\log_{10}{\left(10\left(2x-9\right)\right)}=\log_{10}{\left(\frac{3x+5}{2}\right)}\) - step4: Set the arguments equal: \(10\left(2x-9\right)=\frac{3x+5}{2}\) - step5: Cross multiply: \(10\left(2x-9\right)\times 2=3x+5\) - step6: Simplify the equation: \(20\left(2x-9\right)=3x+5\) - step7: Expand the expression: \(40x-180=3x+5\) - step8: Move the variable to the left side: \(40x-180-3x=5\) - step9: Subtract the terms: \(37x-180=5\) - step10: Move the constant to the right side: \(37x=5+180\) - step11: Add the numbers: \(37x=185\) - step12: Divide both sides: \(\frac{37x}{37}=\frac{185}{37}\) - step13: Divide the numbers: \(x=5\) - step14: Check if the solution is in the defined range: \(x=5,x>\frac{9}{2}\) - step15: Find the intersection: \(x=5\) The solution to the equation \(1+\log (2x-9)=\log (3x+5)-\log 2\) is \(x=5\).