Question

How do you solve for x?: `e^(-2x) = 1/3?`

Answer 1

`x=ln(3)/2~~0.54930614433`

Explanation:

Convert the equation to logarithm form.

`color(white)(XX)e^(-2x)=1/3`

`color(white)(XX)hArrlog_e(1/3)=-2x`

`color(white)(XX)ln(1/3)=-2x`

Isolate x.

`color(white)(XX)(-1/2)[ln(1/3)]=(-1/2)(-2x)`

`color(white)(XX)(-ln(1/3))/2=x`

Simplify.

`color(white)(XX)ln((1/3)^-1)/2=x` Theorem: Logarithm of a Power

`color(white)(XX)color(red)(x=ln(3)/2)`

You can leave the answer at that since you can't really get `ln(3)` without a calculator. The actual answer is around 0.54930614433.