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This one has me stumped. I know that there is no law of exponents explaining what happens for a sum of exponents with the same base, so I tried taking the natural log on both sides and adding the exponents that way as they multiply the same base, but it doesn't seem to be correct and I haven't been able to find a good explanation for how to proceed.

The problem is: 2^(x+2) + 2^(x+1) + 2^(x) = 3/4

If this question is against any of SE's rules I would be more than pleased to know of a good source to learn how to solve these types of problems, or even simply a nudge in the right direction! Thanks.

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write $$2^x\cdot 2^2+2^x\cdot 2+2^x=\frac{3}{4}$$ can you finish?

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  • $\begingroup$ The RHS is $3/4$, correct? $\endgroup$ – Tim Thayer Mar 20 '18 at 18:01
  • $\begingroup$ yes this was a typo $\endgroup$ – Dr. Sonnhard Graubner Mar 20 '18 at 18:02
  • $\begingroup$ I deeply appreciate your help! Thanks for assisting me in fostering greater mathematical intuition $\endgroup$ – Jake Mar 20 '18 at 18:39
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$$ 2^{x+2} + 2^{x+1} + 2^x = 3/4 $$

$$ 4 (2^x) +2(2^x) + 2^x = 3/4 $$

Let $y=2^x$ and substitute to get $$ 7y=3/4 $$

$$ y= \frac{3}{28}$$

$$2^x= \frac{3}{28}$$

$$ x \ln(2) = \ln(\frac{3}{28})$$

Solve for $x$.

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  • $\begingroup$ Thank you, my final answer of -3.222 coincides with the result I was able to reach on a website but without understanding the process. Very appreciative. $\endgroup$ – Jake Mar 20 '18 at 18:41
  • $\begingroup$ Which part you did not get? $\endgroup$ – Mohammad Riazi-Kermani Mar 20 '18 at 18:46
  • $\begingroup$ I meant before understanding the process with the help of your answer, sorry if I was unclear. $\endgroup$ – Jake Mar 21 '18 at 17:11
  • $\begingroup$ No problem. Thanks for the comment. $\endgroup$ – Mohammad Riazi-Kermani Mar 21 '18 at 17:17

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