Find the indicated value of the logarithmic function.
[tex]$\ln \left(e^{-4}\right)$[/tex]
Answer (1)
We are asked to evaluate the natural logarithm of e − 4 .
Recall the property ln ( e x ) = x .
Apply the property to find ln ( e − 4 ) = − 4 .
The final answer is − 4 .
Explanation
Understanding the Problem We are asked to find the value of the natural logarithm of e − 4 . The natural logarithm, denoted as ln ( x ) , is the logarithm to the base e . In other words, ln ( e − 4 ) asks the question: to what power must we raise e to obtain e − 4 ?
Using Logarithmic Properties To evaluate ln ( e − 4 ) , we can use the property of logarithms that states ln ( e x ) = x for any real number x . This property is a direct consequence of the definition of the natural logarithm as the inverse function of the exponential function e x .
Applying the Property Applying this property to the given expression, we have ln ( e − 4 ) = − 4 . This is because e raised to the power of − 4 is equal to e − 4 .
Final Answer Therefore, the value of the logarithmic function ln ( e − 4 ) is simply − 4 .
Examples
Logarithmic functions are incredibly useful in many real-world applications, such as calculating the time it takes for an investment to double at a certain interest rate. For example, if you want to know how long it will take for an investment to double at an annual interest rate that is continuously compounded, you can use the formula t = r l n ( 2 ) , where t is the time in years and r is the interest rate. Logarithms also appear in physics, such as calculating the decay of radioactive materials or measuring the magnitude of earthquakes on the Richter scale.
