Question 4: Find the value of \(x\) in the following equations:
\((iv)\) \(\quad \)\(\left(\frac{1}{27}\right)^{x-6} =27\)
Solution:
\[
\begin{align*}
\left(\frac{1}{27}\right)^{x-6} &=27\\
\left(\frac{1}{3^{3}}\right)^{x-6} &=3^{3}\\
\left( 3^{-3}\right)^{x-6} &=3^{3}\\
( 3)^{-3( x-6)} &=3^{3}\\
3^{-3x+18} &=3^{3}
\end{align*}
\]
Equating bases, we have
\[
\begin{align*}
-3x+18&=3\\
-3x&=3-18\\
-3x&=-15\\
x&=\frac{-15}{-3}\\
x&=5
\end{align*}
\]