Solve the following exponential equation: $$3^{x+2}+3^{x+1} \cdot 3^x+3^{x-1}=120$$
See development and solution
Development:
$$3^{x+2}+3^{x+1} \cdot 3^x+3^{x-1}=120$$
Extracting common factor of $$3^x$$ we get $$$9\cdot3^x+3\cdot3^x+3^x+ \dfrac{3^x}{3} =120$$$ multiplying by $$3$$ one has $$$27\cdot 3^x+9\cdot3^x+3\cdot 3^x+3^x=360 \Rightarrow 3^x(27+9+3+1)=360 \Rightarrow 3^x=\dfrac{360}{40} \Rightarrow$$$ $$$\Rightarrow 3^x=9 \Rightarrow x=2$$$
Solution:
$$x=2$$