# Solving an equation with exponentials

$$2^x+4^x+12=0$$

How exactly am I supposed to solve this? Am I supposed to get $x$ alone or solve it another way?

• Hint: write $u = 2^x$ and find a quadratic in $u$. – Simon S Dec 11 '14 at 3:36
• Assuming that x is a real number, what do you know about any positive number raised to a real power (in this case $2^x$ and $4^x$) can it be negative? – JMoravitz Dec 11 '14 at 3:39
• In the future, please provide a more descriptive title; it makes it easier for users to help you. Also, users typically appreciate a more respectful tone than the imperative "solve this problem!" – Gyu Eun Lee Dec 11 '14 at 4:16
• If this is indeed precalculus . . . did you maybe transmute a - to a +? – imallett Dec 11 '14 at 5:20
• What does "get $x$ alone" mean? – JiK Dec 11 '14 at 10:05

For any $x \in \mathbb{R}$, $2^x > 0$ and $4^x > 0$, therefore

$$2^x + 4^x + 12 > 0 + 0 + 12 = 12 > 0$$

Therefore there is no real solution.

• but no condition is given in the question stating $x\in R$ – Dheeraj Kumar Dec 11 '14 at 3:43
• @DheerajKumar The tag is "algebra precalculus", so what are the odd this is a complex variable exercise? – Timbuc Dec 11 '14 at 3:45
• You are right. But otherwise specified I think it is reasonabe to assume the convention $x$ denotes a member of $\mathbb{R}$ and $z$ denotes a member of $\mathbb{C}$. – Empiricist Dec 11 '14 at 3:45

let $u = 2^x$, then $4^x = (2^2)^x = 2^{2x} = 2^{2x} = (2^x)^2 = u^2$. Thus, $2^x + 4^x + 12 = 0$ becomes, $$u + u^2 + 12 = 0$$ Using the quadratic equation will solve $u$, which is really $2^x$. To solve for the $x$, just take $\log$ on both sides of the solution, then after rearranging, you should be able to solve for $x$.

• With the constraint that $u>0$... – Umberto Dec 11 '14 at 7:11