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I have this question:

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

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

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

$3^{12} = 3^x$

$x = 12$

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Answer:

$$3^{11}+3^{11}+3^{11} = 3^x\implies 3^{10}(3^{1} + 3^{1} + 3^{1}) =3^x$$ $$(3^1 + 3^1+ 3^1) = \frac {3^x} {3^{10}}$$ $$9 = \frac{3^x}{3^{10}}$$ $$3^2 = \frac {3^x}{3^{10}}$$ Hence: $$2 = x -10$$ $$x = 12$$

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  • 2
    $\begingroup$ Ah, one of those self-answered questions? $\endgroup$ – Simply Beautiful Art Jan 31 '17 at 12:47
  • $\begingroup$ Yes, it was meant to be shared ;) $\endgroup$ – Adola Jan 31 '17 at 12:49
  • $\begingroup$ Its good you solved it by yourself. $\endgroup$ – Kanwaljit Singh Jan 31 '17 at 13:00
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More simply:

$$3^{11}+3^{11}+3^{11}=3\times 3^{11}=3^{12}$$

so the answer you are looking for is $x=12$ (because $x\mapsto 3^x$ is injective).

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