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

Answer 1

Eva Adamson

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

Translate the formula into logarithmic form.

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

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

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

Separate x.

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

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

Make it simple.

#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.

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Answer 2

Ethan Adkins

To solve for (x), take the natural logarithm (ln) of both sides of the equation, then isolate (x) by dividing by the coefficient of (x). The solution is (x = \frac{-1}{2} \ln\left(\frac{1}{3}\right)).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.