Given the following logarithmic function below, what is the value of \( f\left(e^{4}\right) \) ? \( f(x)=\ln x \) 4 7 6 5

To find \( f\left(e^{4}\right) \), plug \( e^{4} \) into the function \( f(x) = \ln x \). When you calculate \( f(e^{4}) = \ln(e^{4}) \), you can use the property of logarithms that states \( \ln(a^b) = b \cdot \ln(a) \). Since \( \ln(e) = 1 \), it simplifies to \( 4 \cdot 1 = 4 \). So, the value of \( f(e^{4}) \) is \( 4 \). While logarithms may seem abstract, they have real-world applications in fields like finance and science. For instance, they can be used to model exponential growth, such as population growth or compound interest, allowing us to make predictions and informed decisions based on initial data. When dealing with logarithmic functions, one common mistake is confusing the base when calculating logarithms. Always ensure you know whether you’re working with natural logarithms (base \( e \)) or common logarithms (base 10). Another tip is to remember that logarithms can only be applied to positive numbers—so if you attempt to calculate the logarithm of a negative number, you’ll run into trouble!