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A natural logarithm is denoted by the symbol \( \ln \), which stands for the natural logarithm base \( e \). The constant \( e \) is approximately equal to 2.71828, and it is a fundamental mathematical constant often encountered in calculus and mathematical analysis. The natural logarithm of a number \( x \) is the power to which \( e \) must be raised to obtain the number \( x \). For example, \( \ln(e) = 1 \) because \( e^1 = e \).
Natural logarithms are commonly used in problems involving exponential growth and decay, as well as in calculus due to their unique properties that simplify differentiation and integration.

• \( \ln(ab) = \ln(a) + \ln(b) \)
• \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \)
• \( \ln(a^b) = b \cdot \ln(a) \)

Understanding natural logarithms is crucial for solving logarithmic equations that involve such expressions.

Solving equations, particularly logarithmic ones like \( \ln(2x-4) = \ln(x+6) \), means finding the value of \( x \) that satisfies the equation. For logarithmic equations with the same base on both sides, like in this problem, you can equate the arguments directly. This is because if \( \ln(A) = \ln(B) \), then \( A = B \).When solving such equations, follow these steps:

• First, identify the arguments of the logarithms, \( 2x-4 \) and \( x+6 \), and set them equal to each other.
• Then, rearrange the resulting equation \( 2x - 4 = x + 6 \) to solve for \( x \). In this case, subtracting \( x \) from both sides and then adding 4 results in \( x = 10 \).

Finally, always verify your solution by checking if the original logarithmic expressions are valid at your solution, as logarithms are only defined for positive arguments.

When dealing with logarithmic functions such as \( \ln(x) \), it is essential to understand their domain. The domain of a logarithmic function is the set of all positive real numbers. This is because the logarithm of a non-positive number is not defined in real numbers.
In the context of solving \( \ln(2x-4) = \ln(x+6) \), it is crucial to ensure that both \( 2x-4 \) and \( x+6 \) are positive for the solution \( x \) to be valid.

• For \( 2x - 4 > 0 \), this means \( x > 2 \).
• For \( x + 6 > 0 \), this means \( x > -6 \).

Therefore, \( x = 10 \) satisfies these conditions because it is greater than both 2 and -6. Remember to always check for such constraints to avoid invalid solutions, ensuring that each argument of the logarithm is within the acceptable domain.