# Find the value of $x$ that satisfy the equation: $3^{11}+3^{11}+3^{11} = 3^x$

I have this question:

$$3^{11}+3^{11}+3^{11} = 3^x$$ Find the value of $x$

• $3^{11}+3^{11}+3^{11}=3^{12}$ and the exponential is injective. – Piquito Jan 31 '17 at 12:47

Takking $3^{11}$ common from left hand side,

$3^{11}(1 + 1 + 1) = 3^x$

$3^{11}(3) = 3^x$

$3^{12} = 3^x$

$x = 12$

Or -

$3(3^{11})= 3^x$

$3^{12} = 3^x$

$x = 12$

$$3^{11}+3^{11}+3^{11} = 3^x\implies 3^{10}(3^{1} + 3^{1} + 3^{1}) =3^x$$ $$(3^1 + 3^1+ 3^1) = \frac {3^x} {3^{10}}$$ $$9 = \frac{3^x}{3^{10}}$$ $$3^2 = \frac {3^x}{3^{10}}$$ Hence: $$2 = x -10$$ $$x = 12$$
$$3^{11}+3^{11}+3^{11}=3\times 3^{11}=3^{12}$$
so the answer you are looking for is $x=12$ (because $x\mapsto 3^x$ is injective).