To solve the equation [tex]\( e^{2x} = 50 \)[/tex], follow these steps:
1. Understand the given exponential equation:
[tex]\[ e^{2x} = 50 \][/tex]
2. Take the natural logarithm (ln) of both sides:
[tex]\[ \ln(e^{2x}) = \ln(50) \][/tex]
3. Apply the power rule of logarithms. The power rule states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Using this rule:
[tex]\[ 2x \cdot \ln(e) = \ln(50) \][/tex]
4. Simplify the equation. Note that [tex]\(\ln(e) = 1\)[/tex], so we have:
[tex]\[ 2x = \ln(50) \][/tex]
5. Solve for [tex]\(x\)[/tex]. Isolate [tex]\(x\)[/tex] by dividing both sides by 2:
[tex]\[ x = \frac{\ln(50)}{2} \][/tex]
6. Calculate the value of [tex]\(\ln(50)\)[/tex]. Using a calculator, you find:
[tex]\[ \ln(50) \approx 3.912 \][/tex]
7. Divide this value by 2 to find [tex]\(x\)[/tex]:
[tex]\[ x \approx \frac{3.912}{2} = 1.956 \][/tex]
8. Round the solution to the thousandths place. The value is already in the correct form:
[tex]\[ x \approx 1.956 \][/tex]
Therefore, the solution to the equation [tex]\( e^{2x} = 50 \)[/tex] is:
[tex]\[ x \approx 1.956 \][/tex]
So the correct answer is:
[tex]\[ x = 1.956 \][/tex]
