Step 1 :\(\log (7 x+9) - 1 = \log (x-1)\)
Step 2 :\(\log (7 x+9) - \log 10 = \log (x-1)\)
Step 3 :\(\log \left(\frac{7x+9}{10}\right) = \log (x-1)\)
Step 4 :\(\frac{7x+9}{10} = x-1\)
Step 5 :\(7x + 9 = 10x - 10\)
Step 6 :\(9 = 3x - 10\)
Step 7 :\(19 = 3x\)
Step 8 :\(x = \frac{19}{3}\)
Step 9 :Check: \(\log (7 \cdot \frac{19}{3} + 9) = 1 + \log (\frac{19}{3} - 1)\)
Step 10 :Check: \(\log (44) = 1 + \log (5)\)
Step 11 :\(\boxed{\text{No Solution}}\)