91Ó°ÊÓ

The domain of a logarithmic function is crucial to understand when solving logarithmic equations. For any logarithm, the argument (the value inside the log) must be greater than zero because the logarithm of a non-positive number is undefined.
Specifically, in our sample problem, the equation given is \(6 \ln(2x) = 30\). The argument of the logarithm function here is \(2x\). Therefore, the domain is defined by the inequality \(2x > 0\), which simplifies to \(x > 0\). This means we are only looking for solutions where \(x\) is positive.
Understanding the domain ensures that the solution we derive from the equation is valid and falls within the acceptable range. Any solution that does not satisfy \(x > 0\) must be rejected.

Converting a logarithmic equation into its exponential form is a key step in solving for the variable. The natural logarithm, denoted as \(\ln\), is based on the constant \(e\), which is approximately 2.71828.
For any equation of the form \(\ln(a) = b\), this can be rewritten using exponential form as \(a = e^b\).
In our exercise, we had \(\ln(2x) = 5\). By converting to exponential form, we translated it to \(2x = e^5\).
This step is fundamental because it shifts the problem from a logarithmic context to an exponential one, which can often be more straightforward to solve.

After solving the logarithmic equation and ensuring that the solution is within the domain, we often seek a decimal approximation for practical use.
In our case, after solving for \(x\), we found \(x = \frac{e^5}{2}\). To find a useful decimal value, a calculator or computational tool is used.
It's important to know how to correctly approximate this value to the required decimal places. In this instance, we rounded \(x\) to two decimal places resulting in \(x \approx 67.67\).
Decimal approximation makes it easier to interpret the results and apply them in real-world situations, providing a clear image of the result's magnitude.

The process of solving for \(x\) in logarithmic equations involves a few methodical steps. First, isolate the logarithmic term to simplify your equation.
Once the equation is simplified, move the equation into exponential form, as this makes it easier to manipulate algebraically.
In the given task, once we had \(\ln(2x) = 5\), we turned it into \(2x = e^5\). The next step is to solve algebraically for \(x\).
This involves dividing both sides of the equation by 2, giving \(x = \frac{e^5}{2}\). Finally, verify your solution meets the domain's requirements before using any tools for a more precise decimal approximation.
Each step is essential to ensure accuracy and relevance of the answer obtained from the logarithmic equation.