How do you solve for x?: $e^{- 2 x} = \frac{1}{3} ?$ $e^(-2x) = 1/3?$

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Answer 1 · 2015-10-23T16:29:50

$x = \frac{ln (3)}{2} \approx 0.54930614433$ $x=ln(3)/2~~0.54930614433$

Convert the equation to logarithm form.

$X X e^{- 2 x} = \frac{1}{3}$ $color(white)(XX)e^(-2x)=1/3$

$X X \Leftrightarrow {log}_{e} (\frac{1}{3}) = - 2 x$ $color(white)(XX)hArrlog_e(1/3)=-2x$

$X X ln (\frac{1}{3}) = - 2 x$ $color(white)(XX)ln(1/3)=-2x$

Isolate x.

$X X (- \frac{1}{2}) [ln (\frac{1}{3})] = (- \frac{1}{2}) (- 2 x)$ $color(white)(XX)(-1/2)[ln(1/3)]=(-1/2)(-2x)$

$X X \frac{- ln (\frac{1}{3})}{2} = x$ $color(white)(XX)(-ln(1/3))/2=x$

Simplify.

$X X \frac{ln ({(\frac{1}{3})}^{- 1})}{2} = x$ $color(white)(XX)ln((1/3)^-1)/2=x$ Theorem: Logarithm of a Power

$X X x = \frac{ln (3)}{2}$ $color(white)(XX)color(red)(x=ln(3)/2)$

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

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