13. Which of the following is equivalent to \( e^{4 x}=2981 ? \) a. \( x=\frac{\ln 2981}{4} \) b. \( x=\frac{4}{\ln 2981} \) c. \( x=\frac{\ln 4}{2981} \) d. \( x=\frac{2981}{\ln 4} \)

To solve for \( x \) in the equation \( e^{4 x} = 2981 \), we first take the natural logarithm of both sides. This gives us \( 4x = \ln(2981) \). Dividing both sides by 4, we have \( x = \frac{\ln(2981)}{4} \). So, the correct answer is option a: \( x=\frac{\ln 2981}{4} \). Understanding exponents and logarithms can be traced back to ancient civilizations, where the need to solve practical problems, such as trade and land measurement, led to early forms of these mathematical concepts. Fast forward to today, and logarithms are essential in fields like electronics and information theory, making them not just a relic of history but a crucial tool in modern science and technology! When working with equations involving exponentials and logarithms, one common mistake is neglecting to apply the logarithm correctly or forgetting to check if the base of the logarithm matches the one in the exponent. Remember, it’s all about diligence! Always double-check your steps and ensure you are expressing the equations accurately to avoid mix-ups, especially with coefficient placements like that pesky 4 in this case.