Question

solve for x in the equation below. round your answer to the nearest hundredth. do not round any intermediate computations. $e^{-7x}=3$

Explanation:

Step1: Take natural log of both sides

To solve for \( x \) in the equation \( e^{-7x} = 3 \), we start by taking the natural logarithm (\(\ln\)) of both sides. Recall that \(\ln(e^a)=a\) due to the inverse relationship between the exponential function \( e^x \) and the natural logarithm function \(\ln(x)\). So, applying \(\ln\) to both sides gives:

$$ \ln(e^{-7x})=\ln(3) $$

Using the property \(\ln(e^a) = a\), the left - hand side simplifies to \(-7x\). So we have the equation:

$$ -7x=\ln(3) $$

Step2: Solve for x

Now, we solve for \( x \) by dividing both sides of the equation \(-7x=\ln(3)\) by \(-7\). We know that \(\ln(3)\approx1.098612\). So,

$$ x = \frac{\ln(3)}{-7} $$

Substituting the approximate value of \(\ln(3)\) into the formula, we get:

$$ x=\frac{1.098612}{-7}\approx - 0.1569446 $$

Rounding this value to the nearest hundredth (two decimal places), we look at the third decimal place. The third decimal place is 6, which is greater than or equal to 5. So we round up the second decimal place.

Answer:

\(x\approx - 0.16\)