Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expressions. give the exact answer. \\(\log(7x - 2) = \log(x + 2) + \log 8\\) solve the equation to find the solution set. select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. the solution set is \\(\boxed{}\\). (simplify your answer. use a comma to separate answers as needed.) \\(\bigcirc\\) b. there are infinitely many solutions. \\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Apply log property

Using the property \(\log a+\log b = \log(ab)\), the right - hand side of the equation \(\log(7x - 2)=\log(x + 2)+\log8\) becomes \(\log[8(x + 2)]\). So the equation is \(\log(7x-2)=\log(8x + 16)\).

Step2: Set arguments equal

If \(\log a=\log b\), then \(a = b\) (for \(a>0,b>0\)). So we set \(7x-2=8x + 16\).

Step3: Solve for x

Subtract \(7x\) from both sides: \(7x-2-7x=8x + 16-7x\), which simplifies to \(- 2=x + 16\). Then subtract 16 from both sides: \(x=-2 - 16=-18\).

Step4: Check domain

For \(\log(7x - 2)\) to be defined, \(7x-2>0\Rightarrow x>\frac{2}{7}\). For \(\log(x + 2)\) to be defined, \(x + 2>0\Rightarrow x>-2\). For \(\log8\), it is defined for all real numbers. But our solution \(x = - 18\) does not satisfy \(x>\frac{2}{7}\) (since \(-18<\frac{2}{7}\)). So there is no solution.

Answer:

C. There is no solution.