Question

solve the equation. give the solution in exact form.\\(\log_{4}(x + 7)(x - 5) = 3\\)\
rewrite the given equation without logarithms. do not solve for x.\
\\(\square\\)\
select the correct choice below and, if necessary, fill in the answer box to complete your choice.\
\\(\bigcirc\\) a. the solution set is \\(\\{\square\\}\\).\
\\(\quad\\) (type an exact solution, using radicals as needed. use a comma to separate answers as needed.)\
\\(\bigcirc\\) b. the solution is the empty set.

Explanation:

Step1: Recall the logarithm definition

For a logarithmic equation \(\log_b(a) = c\), it can be rewritten in exponential form as \(b^c = a\) (where \(b>0\), \(b
eq1\), and \(a>0\)). Here, \(b = 4\), \(c = 3\), and \(a=(x + 7)(x - 5)\).

Step2: Apply the definition

Using the definition, we rewrite \(\log_{4}[(x + 7)(x - 5)] = 3\) as \(4^3=(x + 7)(x - 5)\).

Step3: Simplify the exponential term

Calculate \(4^3\), which is \(4\times4\times4 = 64\). So the equation without logarithms is \((x + 7)(x - 5)=64\).

Step4: Solve the quadratic equation

First, expand the left - hand side: \(x^{2}-5x + 7x-35 = 64\).
Simplify to get \(x^{2}+2x-35 - 64=0\), which is \(x^{2}+2x - 99 = 0\).
We can solve this quadratic equation using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a = 1\), \(b = 2\), and \(c=-99\).
Calculate the discriminant \(\Delta=b^{2}-4ac=(2)^{2}-4\times1\times(-99)=4 + 396 = 400\).
Then \(x=\frac{-2\pm\sqrt{400}}{2}=\frac{-2\pm20}{2}\).
We have two solutions:
\(x_1=\frac{-2 + 20}{2}=\frac{18}{2}=9\) and \(x_2=\frac{-2-20}{2}=\frac{-22}{2}=-11\).
Now we need to check the domain of the original logarithmic function. The argument of the logarithm \((x + 7)(x - 5)>0\).
For \(x = 9\): \((9 + 7)(9 - 5)=16\times4 = 64>0\), so \(x = 9\) is valid.
For \(x=-11\): \((-11 + 7)(-11 - 5)=(-4)\times(-16)=64>0\), so \(x=-11\) is also valid.

Answer:

A. The solution set is \(\{ - 11,9\}\)