Solve each equation below. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places.
(1) (cuberoot{2})^(2 - x) = 2^(x^2)
(2) e^(x + 3) = pi^(x)
$\displaystyle (\sqrt[3]{2})^{2-x} = 2^{x^2}$
$\displaystyle (\sqrt[3]{2})^{2-x} = [(\sqrt[3]{2})^3]^{x^2}$
$\displaystyle (\sqrt[3]{2})^{2-x} = (\sqrt[3]{2})^{3x^2}$
so ... what can you say about $\displaystyle 2-x$ and $\displaystyle 3x^2$ ?
$\displaystyle e^{x+3} = \pi^x$
$\displaystyle \ln(e^{x+3}) = \ln(\pi^x)$
$\displaystyle x+3 = x\ln{\pi}$
$\displaystyle 3 = x\ln{\pi} - x$
$\displaystyle 3 = x(\ln{\pi} - 1)$
$\displaystyle \frac{3}{\ln{\pi} - 1} = x$