Question 4: Find the value of \(x\) in the following equations:
\((vi)\) \(\quad\) \(\log_{2} \ ( x+1) -\log_{2} \ ( x-4) =2 \)
Solution:
\[\log_{2}( x+1) -\log_{2}( x-4) =2\]
\[
\begin{align*}
\log_{2}\left(\frac{x+1}{x-4}\right) &=2\\
\frac{x+1}{x-4} &=2^{2}\\
\frac{x+1}{x-4} &=4\\
x+1&=4( x-4)\\
x+1&=4x-16\\
x-4x&=-16-1\\
-3x&=-17\\
x&=\frac{-17}{-3}\\
x&=\frac{17}{3}\\
x&=5\frac{2}{3}
\end{align*}
\]