Question

Apply the inverse properties of \(\ln x\) and \(e^{x}\) to simplify the given expression. $$ -1+\ln e^{2 x} $$

Short answer

The simplified form of the expression \(-1+\ln e^{2 x}\) is \(2x - 1\).

Step-by-step solution

Identify Functions

In the given expression, \(-1+\ln e^{2x}\), there is an application of the natural logarithm to the exponential function \(e^{2x}\). Given that natural logarithm and the function with base e are inverse functions, they cancel each other out, leaving only the argument of the exponential function.

Apply Inverse Property

Applying the inverse property of natural logarithm and \(e^{x}\), we get the argument of the exponential function. In this case it leaves \(2x\). So the expression becomes \(-1+ 2x\).

Reorder the Expression

It is common to write the expression in order of power which gives a more simplified version of our expression. Write the terms in order of descending power of x. Therefore, our final simplified expression is \(2x - 1\).