91Ó°ÊÓ

A natural logarithm, denoted as \(\text{ln}\), is a logarithm to the base of the mathematical constant \(e\) (approximately 2.718). It is a way to determine the power to which the base \(e\) must be raised to produce a given number. Natural logarithms are commonly used in various fields such as calculus, physics, and engineering due to their properties and the ease of deriving functions involving exponentials.

For instance, \(\text{ln}(e^x) = x\), as \(e\) raised to the power of \(x\) yields \(e^x\). Similarly, \(\text{ln}(1) = 0\) because \(e^0 = 1\). Natural logarithms help in transforming multiplicative relationships into additive ones, simplifying the processes of solving exponential and logarithmic equations.

In our exercise, we are given \(\text{ln} (3x) = \text{ln} 5\), leading us to apply the properties of natural logarithms.

Understanding the equality property of logarithms is crucial. This property tells us that if two logarithmic expressions with the same base are equal, then their corresponding arguments are also equal.

Specifically, if \(\text{ln} (A) = \text{ln} (B)\), then \(A = B\). This principle simplifies solving equations involving logarithms by allowing us to bypass the logarithmic form and directly compare the arguments inside the logarithms.

Let's apply this property step-by-step to our equation. We start with: \( \text{ln} (3x) = \text{ln} (5) \)

Using the equality property, we equate the arguments directly: \( 3x = 5 \). At this point, the logarithmic component has been eliminated, simplifying the solving process.

To isolate the variable, we need to get \(\text{x}\) alone on one side of the equation. This typically involves applying inverse operations to both sides of the equation based on the operations currently applied to the variable.

Let's revisit the equation obtained after applying the equality property: \( 3x = 5 \).

Here, \( x \) is being multiplied by 3. To isolate \(\text{x}\), we divide both sides of the equation by 3:

\[ x = \frac{5}{3} \]

This operation effectively reverses the multiplication, leaving \( x \) isolated. Now \( x \) equals \( \frac{5}{3} \), which simplifies the process of finding the solution.

As a final step, we confirm that \(\frac{5}{3}\) cannot be simplified further, marking the completion of the exercise.