solve the equation
ln(2+e^-x)=2 its e to the power -x
answer correct to 2 dp
This equation can be solved exactly:
$\displaystyle \ln(2+e^{-x})=2$
$\displaystyle \ln \left(\dfrac{2e^x+1}{e^x} \right)=2$
$\displaystyle \ln (2e^x+1) + \ln(e^{-x})=2$
$\displaystyle \ln (2e^x+1) =x + 2$
$\displaystyle 2e^x+1 =e^{x + 2} = e^2 \cdot e^x$
$\displaystyle 1 = e^2 \cdot e^x - 2e^x = e^x(e^2-2)$
$\displaystyle \dfrac1{e^2-2} = e^x$
$\displaystyle -\ln(e^2-2) = x$