What is the value of [tex]$e^{\ln 7 x}$[/tex]?
Answer (1)
Recognize that e and ln are inverse functions.
Apply the property e l n u = u .
Simplify the expression e l n 7 x to 7 x .
The value of the expression is 7 x .
Explanation
Understanding the Expression We are asked to simplify the expression e l n 7 x . This expression involves an exponential function with base e and a natural logarithm in the exponent.
Using Inverse Property To simplify this expression, we need to recall the property that the exponential function and the natural logarithm are inverse functions of each other. Specifically, e l n u = u for any 0"> u > 0 .
Simplifying the Expression Applying this property to our expression, we have e l n 7 x = 7 x . This holds true as long as 0"> 7 x > 0 , which means 0"> x > 0 .
Final Answer Therefore, the simplified expression is 7 x .
Examples
In physics, exponential functions and logarithms are used to model various phenomena, such as radioactive decay. If the amount of a radioactive substance at time t is given by N ( t ) = N 0 e − k t , where N 0 is the initial amount and k is a constant, then the time it takes for the substance to decay to a certain level can be found using logarithms. For example, if we want to find the time it takes for the substance to decay to half its initial amount, we would solve for t in the equation 2 1 N 0 = N 0 e − k t , which involves taking the natural logarithm of both sides.
