Question

which statement is true about the potential solution ( x = -5 ) of the equation ( log_{4}(2 - x) = log_{4}(-5x - 18) )?
this number is a true solution of the original equation.
this number is not a true solution because it is negative.
this number is not a true solution because ( 2 - x ) is negative.
this number is not a true solution because ( -5x - 18 ) is negative.

Explanation:

Step1: Recall the domain of logarithmic functions

For a logarithmic function \(\log_b(y)\), the argument \(y\) must be positive, i.e., \(y>0\). So we need to check the arguments of both logarithms in the equation \(\log_4(2 - x)=\log_4(-5x - 18)\) when \(x = - 5\).

Step2: Check the argument of the first logarithm (\(2 - x\))

Substitute \(x=-5\) into \(2 - x\):
\(2-(-5)=2 + 5=7\), which is positive.

Step3: Check the argument of the second logarithm (\(-5x - 18\))

Substitute \(x = - 5\) into \(-5x-18\):
\(-5\times(-5)-18=25 - 18 = 7\), which is positive.

Step4: Solve the original equation

If \(\log_b(m)=\log_b(n)\), then \(m = n\) (for \(m>0\), \(n>0\) and \(b>0,b
eq1\)).
Set \(2 - x=-5x - 18\)
\(2+18=-5x + x\)
\(20=-4x\)
\(x=-5\)
Since when \(x = - 5\), both \(2 - x\) and \(-5x - 18\) are positive (equal to 7), \(x=-5\) is a valid solution.

Answer:

This number is a true solution of the original equation.