Question

8^(sqrt(x)) = 4^(sqrt(x)) + 1

find x

Answer 1

\[8^{\sqrt{x}}=4^{\sqrt{x}}+1\]
Let's find \[y=2^\sqrt{x}\] first
\[y^3=y^2+1 \rightarrow y^3-y^2-1=0\]

Answer 2

http://www.wolframalpha.com/input/?i=8%5Esqrt%28x%29%3D4%5Esqrt%28x%29%2B1

Answer 3

Let u = 2^sqrt(x). Then notice that u^2 = 4^sqrt(x) and u^3 = 8^sqrt(x)

So u^3 = u^2 + 1 ....

u^3 - u^2 - 1 = 0

Answer 4

The unfortunate thing now is that this equation looks simple but doesn't have a particularly nice solution.

Answer 5

u^2 = 2^(2(sqrt(x))