If \( e^{5 x}=22 \), then \( x=\square \)

To solve the equation \( e^{5x} = 22 \), you can take the natural logarithm of both sides. This gives: \[ 5x = \ln(22) \] Next, divide both sides by 5: \[ x = \frac{\ln(22)}{5} \] Using a calculator, you can find that \( \ln(22) \approx 3.091 \). Therefore: \[ x \approx \frac{3.091}{5} \approx 0.6182 \] So, \( x \approx 0.6182 \). If you want to verify your answer, plug \( x \) back into the original equation: calculate \( e^{5 \cdot 0.6182} \) and see if it equals approximately 22. It's a helpful way to confirm you followed the right steps! Mathematics has a fun and quirky history! The number \( e \) is named after the Swiss mathematician Leonhard Euler, who introduced it in the 18th century. This constant is not just key in calculus; it also pops up in scenarios ranging from finance to statistics, connecting growth models and decay processes. Isn’t it fascinating how one number can play such diverse roles across different fields? Speaking of real-world scenarios, you can encounter exponential equations like this one in finance when dealing with compound interest or in population studies predicting growth over time. For instance, if you invest money in a savings account with compounded interest, the formula involves \( e \). Understanding how to manipulate equations like this helps in making informed financial decisions, illustrating the practicality of math in everyday life!