Question

find the potential solution to the equation \\(\log_{4}(2 - x) = \log_{4}(-5x - 18)\\).
\\(x = \square\\)

Explanation:

Step1: Use the property of logarithms

If \(\log_b a=\log_b c\), then \(a = c\) (for \(b>0,b
eq1,a>0,c>0\)). So we set the arguments equal:
\(2 - x=-5x - 18\)

Step2: Solve for \(x\)

Add \(5x\) to both sides:
\(2 - x+5x=-5x - 18+5x\)
\(2 + 4x=-18\)
Subtract 2 from both sides:
\(4x=-18 - 2\)
\(4x=-20\)
Divide both sides by 4:
\(x=\frac{-20}{4}=-5\)
Now we check the domain of the logarithms. For \(\log_4(2 - x)\), \(2-(-5)=7>0\). For \(\log_4(-5x - 18)\), \(-5\times(-5)-18 = 25 - 18 = 7>0\). So \(x = - 5\) is valid.

Answer:

\(-5\)