Solving an exponential equation by reducing to an equal base

Solve the following exponential equation: $3^{x+2}+3^{x+1} \cdot 3^x+3^{x-1}=120$

$$3^{x+2}+3^{x+1} \cdot 3^x+3^{x-1}=120$$

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 $$27\cdot 3^x+9\cdot3^x+3\cdot 3^x+3^x=360 \Rightarrow 3^x(27+9+3+1)=360 \Rightarrow 3^x=\dfrac{360}{40} \Rightarrow$$ $$\Rightarrow 3^x=9 \Rightarrow x=2$$

$x=2$

Back to topic