Problem 95 - College Algebra Essentials

Question

Evaluate or simplify each expression without using a calculator. $$\ln e^{9 x}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(9x\).

Step 1: Recognizing and Interpreting the Mathematical Properties

In this exercise, a simplification has to be done using the properties of the natural logarithm. It's pivotal to note that the natural logarithm of Euler's number \(e\) raised to any power \(x\) simplifies to just \(x\). This implies that if we have \(\ln e^{kx}\), it can be simplified to \(kx\). This property arises from the fact that \(e\) and \(\ln\) are inverse functions.

Step 2: Applying the Logarithmic Properties

In the given expression, \(\ln e^{9x}\), \(e\) is raised to the power \(9x\). Therefore, applying the natural logarithm property, it simplifies to \(9x\).