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The natural logarithm is a special type of logarithm that uses the base \(e\), where \(e\) is the mathematical constant approximately equal to 2.71828. It is denoted as \(\ln\), which stands for "natural logarithm." The natural logarithm is particularly useful in mathematics because it appears frequently in equations describing growth processes, such as exponential growth and decay. It helps in transforming exponential equations into linear equations, making them easier to solve. In the original exercise, the equation \(\ln(x-3) = 5\) involves a natural logarithm. To solve it, you take the exponential of both sides to remove the logarithmic function, leading to \(x-3 = e^5\). This allows us to isolate \(x\) and find its value by solving the resulting equation.

The exponential function is a mathematical function represented as \(e^x\), where \(e\) is the base of natural logarithms. This function is crucial in applications such as compound interest, population growth, and radioactive decay, because it models growth that increases at a constant percentage rate. In solving logarithmic equations, the exponential function plays a critical role in "undoing" the natural logarithm. In part (a) of the exercise, after isolating the term inside the logarithm, we use the property that the natural logarithm and the exponential function are inverses. By converting \(\ln(x-3) = 5\) into \(x - 3 = e^5\), we essentially "expose" the value of \(x\), enabling us to solve for it easily.

The logarithm product rule is a helpful property that allows us to simplify logarithmic expressions that involve the logarithm of a product. It states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, it is defined as \(\log_b(mn) = \log_b(m) + \log_b(n)\). In part (b) of the exercise, we encounter the expression \(\ln(x+2) + \ln(x-2)\). By using the product rule, this can be combined into a single logarithm: \(\ln((x+2)(x-2))\). This simplification is particularly valuable because it allows us to transform the problem into a more manageable form and eventually solve it by converting it into an exponential equation.

The logarithmic division rule, another essential property, allows us to handle logarithms of quotients effortlessly. It states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\). In part (c) of the exercise, we find \(\log_3(x^2) - \log_3(2x)\). Using the division rule, this can be expressed as a single logarithm: \(\log_3\left(\frac{x^2}{2x}\right)\). Simplifying the expression inside the logarithm leads to \(\log_3\left(\frac{x}{2}\right)\). This simplification helps in converting the logarithmic equation into an exponential one: \(\frac{x}{2} = 9\). It simplifies the process of solving for \(x\) by avoiding complexities that might arise from handling multiple separate logarithmic expressions.