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Logarithms are a powerful tool in mathematics for transforming multiplicative relationships into additive ones. In particular, a **common logarithm** is a logarithm with base 10. It is widely used and is often simply written as \( \log \) without any base, because base 10 is so commonly used. This makes common logarithms extremely useful for calculations and modeling.To understand this better, let’s look at how they function in equations. Common logarithms simplify operations by transforming products and quotients into sums and differences, thanks to identities like:

• \( \log(a \cdot b) = \log(a) + \log(b) \)
• \( \log(a / b) = \log(a) - \log(b) \)

In our original exercise, these identities help us simplify an equation by reducing it from a difference of logarithms to a single logarithm, \( \log\left(\frac{x-4}{3x-10}\right) \). These transformations are not just simplifications; they're essential for the solution because they allow us to compare the arguments (the values inside the logs directly) when both sides of the equation are logarithms.

Quadratic equations appear frequently in algebra. These equations take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. They can typically be solved by factoring, using the quadratic formula, or by completing the square. In our exercise, after applying rules of logarithms and simplifying, we arrive at a quadratic equation: \( x^2 - 7x + 10 = 0 \). To solve by factoring, we look for two numbers that multiply to the constant term, 10, and add to the linear coefficient term, -7. These numbers are -5 and -2, hence the factors are \((x-5)(x-2)\).Each factor provides a potential solution by setting them to zero:

• \( x - 5 = 0\) which gives \( x = 5 \).
• \( x - 2 = 0\) which gives \( x = 2 \).

It’s crucial to factor effectively, as each factor corresponds to a potential solution for the original equation. However, remember to check these potential solutions against any domain restrictions.

When solving equations, particularly those involving logarithms, checking solutions against domain restrictions is critical. Domain restrictions are values that a variable cannot take for the function or expression to be defined. In logarithmic expressions, the argument must be greater than zero since logs of non-positive numbers are undefined.In the problem, the conditions \( x - 4 > 0 \) and \( 3x - 10 > 0 \) stem from the original terms in the logarithms, resulting in:

• \( x > 4 \)
• \( x > \frac{10}{3} \approx 3.33 \)

Each of these inequalities limits acceptable solutions to ensure that both original log terms are defined as positive values. Thus, any potential solution from the quadratic equation must meet all domain restrictions.Upon checking, \( x = 5 \) satisfies both restrictions, whereas \( x = 2 \) does not meet the requirement of \( x > 4 \). Therefore, we reject \( x = 2 \) as a valid solution, leaving \( x = 5 \) as the only solution aligned with both the arithmetic and the domain constraints.