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To understand the natural logarithm, let's first talk about logarithms in general. A logarithm essentially helps us find the power to which a number, called the base, must be raised to produce a given number. In the case of natural logarithms, the base used is Euler's number, denoted as \( e \), which is approximately equal to 2.71828.

Natural logarithms are expressed using the notation \( \ln \). It's important in exponential and logarithmic equations because of its relationship with the exponential function.\( \ln(e^{x}) \) directly helps us find \( x \) when given \( e^{x} \), due to its property of being the inverse of the exponential function.

• The natural logarithm of \( e \) itself is 1, i.e., \( \ln(e) = 1 \)
• Natural logarithms are commonly used in mathematics for growth and decay problems involving exponential functions.

Isolating the expression is a crucial step when dealing with equations, especially when it involves functions like exponential or logarithmic terms. The purpose of this step is to simplify the equation such that the expression you need to solve is by itself on one side of the equation. This simplifies the process of solving for the variable.

In our example, the original equation was \( 3e^{x} = 10 \). Here, the expression \( e^{x} \) was initially multiplied by 3. To isolate \( e^{x} \), we divided both sides by 3, resulting in \( e^{x} = \frac{10}{3} \). This isolated expression then becomes much easier to handle, particularly when the next step involves applying another function such as the natural logarithm.

Evaluating a logarithm involves calculating its numerical value, often using a calculator. For natural logarithms, when you reach this step, you have already simplified the equation to a point where \( x = \ln(\text{some expression}) \). This was the case with \( \ln\left(\frac{10}{3}\right) \).

At this point, the calculator is your best friend in obtaining an accurate numeric solution, especially when decimal precision is required. The capability of a calculator simplifies the process, giving results rounded to a specific decimal place—in this situation, to four decimal places, resulting in \( x \approx 1.2039 \). Ensuring accuracy in this step greatly affects the final outcome of solving exponential equations.