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find a way to solve for x: $$ 2^x = x + 5$$

You can easily see one of the values is $3$. If you plot this in a graph, with $y=2^x$ and $y=x+5$, you'll see it has 2 values. If, in a calculator like casio fx-991E series, PLUS perhaps, you plunk in $2^x=x+5$ and hit solve (shift + calc), you'll get the other value, as the calculator solves from left to right.

But the thing is to find a manual, universal way to solve it

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observe that for $2^x\gt x+5$ for$x\ge 4$

which follows by induction

again $2^x\lt x+5$ for $x\lt 2$

which again follows by induction

so $2\lt x\lt 4$

i think it would be helpful!!!

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  • $\begingroup$ It's not true that $2^x \leq x + 5$ for all $x \leq 2$. As $x \to -\infty$, $2^x \to 0$ but $x + 4 \to -\infty$. $\endgroup$ – Michael Seifert Dec 22 '16 at 13:51
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    $\begingroup$ induction with real numbers? $\endgroup$ – Dr. Sonnhard Graubner Dec 22 '16 at 13:56
  • $\begingroup$ thanks i had mistypped that!!! $\endgroup$ – Blaise Thunderstorm Dec 22 '16 at 13:56
  • $\begingroup$ you can prove it very easily $\endgroup$ – Blaise Thunderstorm Dec 22 '16 at 13:57
  • $\begingroup$ observe decrease & increase in both sides $\endgroup$ – Blaise Thunderstorm Dec 22 '16 at 13:58

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