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In mathematical terms, a natural logarithm is a logarithm to the base of the constant \( e \), which is approximately 2.718. Often noted as \( \ln \), natural logarithms are used to solve equations involving exponential growth or decay, among other applications. When you encounter an equation with a natural logarithm, such as \( \ln 4r^2 = 3 \), the first objective is to eliminate the logarithm to simplify the equation. This involves using the fundamental property that if \( \ln a = b \), then \( a = e^b \). This allows us to convert the logarithmic equation into an exponential form, making it easier to handle. In our case, we reframe \( \ln 4r^2 = 3 \) as \( 4r^2 = e^3 \).Understanding this transformation is crucial because it delineates the pathway to isolating the variable, which is our next step.

Isolating the variable essentially involves getting the variable by itself on one side of the equation to make it easier to solve. With the equation \( 4r^2 = e^3 \) from our natural logarithm transformation, we see the variable in question is \( r \). To isolate \( r \), begin by dividing both sides of the equation by 4, giving us \( r^2 = \frac{e^3}{4} \). Next, to get \( r \) by itself, we take the square root of both sides. Remember that when taking the square root, we consider both the positive and negative roots. Thus, \( r = \sqrt{\frac{e^3}{4}} \) and \( r = -\sqrt{\frac{e^3}{4}} \).This step is vital because it leads to isolating the variable completely, resulting in possible solutions for \( r \). It is also essential to simplify the equation as much as possible to make sure the solutions are accurate.

Once the variable has been isolated and solutions proposed, it is imperative to check these solutions by substituting them back into the original equation. This ensures the solutions are correct and satisfy the given equation.For our equation, we proposed two solutions: \( r = 2.74 \) and \( r = -2.74 \). Plugging these back into the original equation \( \ln 4r^2 = 3 \), we compute the left side for each value of \( r \):

• For \( r = 2.74 \), substitute into the equation: \( 4(2.74)^2 \) and check if this equals \( e^3 \).

• The same process is done for \( r = -2.74 \).

When both solutions lead to the result \( \ln = 3 \), we confirm that the solutions meet the original condition, verifying their correctness.Checking a solution is a crucial step in solving equations as it tests the validity of the outcomes and gives confidence in their accuracy.