To solve the equation [tex]\( e^{2x} = 50 \)[/tex] for [tex]\( x \)[/tex], we can follow these steps:
1. Take the natural logarithm (ln) of both sides of the equation:
[tex]\[
\ln(e^{2x}) = \ln(50)
\][/tex]
2. Simplify the left side using the property of logarithms [tex]\( \ln(e^y) = y \)[/tex]:
[tex]\[
2x = \ln(50)
\][/tex]
3. Solve for [tex]\( x \)[/tex] by isolating it:
[tex]\[
x = \frac{\ln(50)}{2}
\][/tex]
4. Compute the value of [tex]\( x \)[/tex]:
The natural logarithm of 50, [tex]\(\ln(50)\)[/tex], is a number that we can find using a calculator or logarithm table. Once we find [tex]\(\ln(50)\)[/tex], we divide it by 2.
5. Round the result to the thousandths place:
After performing the division, we find the value of [tex]\( x \)[/tex] to be approximately 1.956 when rounded to the nearest thousandth.
Thus, the solution to the equation [tex]\( e^{2x} = 50 \)[/tex] is [tex]\( x \approx 1.956 \)[/tex] when rounded to the thousandths place.
