What is the value of $\ln e^4 $?
Answer (1)
The problem asks to find the value of ln e 4 .
Recall the property of logarithms: ln e x = x .
Apply the property to the expression: ln e 4 = 4 .
The value of ln e 4 is 4 .
Explanation
Understanding the Problem We are asked to find the value of ln e 4 . The natural logarithm, denoted as ln , is the logarithm to the base e . We need to use the property of logarithms that ln e x = x .
Applying the Logarithm Property Using the property of logarithms that ln e x = x , we can apply this to the given expression ln e 4 .
Finding the Value Therefore, ln e 4 = 4 .
Examples
Logarithms are used to simplify calculations in various fields such as physics, engineering, and finance. For example, in finance, the continuously compounded interest formula involves the exponential function and its inverse, the natural logarithm. If you invest a principal amount P at an annual interest rate r compounded continuously, the amount A after t years is given by A = P e r t . To find the time it takes for the investment to double, you would solve for t in the equation 2 P = P e r t , which simplifies to 2 = e r t . Taking the natural logarithm of both sides gives ln 2 = r t , so t = r l n 2 . This shows how logarithms are used to solve exponential equations in real-world applications.
