Question 33, 3.4.67
HW Score: $64.63 \%, 26.5$ of 41 points
Points: 0 of 1
Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expressions. Give an exact answer.
\[
\log _{13} x+\log _{13}(12 x-1)=1
\]
Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Answer
\(\boxed{x = \frac{13}{12}}\)
Step 1 ：\(\log _{13} (x(12x-1))=1\)
Step 2 ：\(13^1 = x(12x-1)\)
Step 3 ：\(13 = 12x^2 - x\)
Step 4 ：\(12x^2 - x - 13 = 0\)
Step 5 ：\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Step 6 ：\(x = \frac{1 \pm \sqrt{(-1)^2 - 4*12*(-13)}}{2*12}\)
Step 7 ：\(x = \frac{1 + \sqrt{625}}{24} = \frac{26}{24} = \frac{13}{12}\)
Step 8 ：\(x = \frac{1 - \sqrt{625}}{24} = -\frac{24}{24} = -1\)
Step 9 ：\(\boxed{x = \frac{13}{12}}\)
