# Math Help - exponential equation help pls

1. ## exponential equation help pls

solve the equation

ln(2+e^-x)=2 its e to the power -x

2. Hi vserian

Hint :

$\log_a b = c$

$b = a^c$

3. Originally Posted by songoku
Hi vserian

Hint :

$\log_a b = c$

$b = a^c$
And in the OP's case a = e

Also $a^{log_a(b)} = b$

4. Originally Posted by vserian
solve the equation

ln(2+e^-x)=2 its e to the power -x

This equation can be solved exactly:

$\ln(2+e^{-x})=2$

$\ln \left(\dfrac{2e^x+1}{e^x} \right)=2$

$\ln (2e^x+1) + \ln(e^{-x})=2$

$\ln (2e^x+1) =x + 2$

$2e^x+1 =e^{x + 2} = e^2 \cdot e^x$

$1 = e^2 \cdot e^x - 2e^x = e^x(e^2-2)$

$\dfrac1{e^2-2} = e^x$

$-\ln(e^2-2) = x$