91Ó°ÊÓ

Logarithmic identities are crucial to understand when solving equations involving logarithms, as they allow us to simplify and manipulate expressions. One key identity is

• \(\log a - \log b = \log\left(\frac{a}{b}\right)\)

This identity helps us to combine or break down logarithmic terms as needed. In the exercise, this identity is used to rewrite the right-hand side of the equation: \(1 - \log(x-2) = \log(10) - \log(x-2)\) This turns the expression into \(\log\left(\frac{10}{x-2}\right)\), making it easier to equate with the left side of the equation, \(\log(x+3)\).
This setting of logs equal to each other leverages another fundamental property of logarithms, which is that if \(\log a = \log b\), then a must equal b. This simplifies our equation further and allows us to proceed with linear and quadratic solutions.

In many mathematical equations, including those involving logarithms, solutions sometimes lead to quadratic equations. A quadratic equation takes the form \((ax^2 + bx + c = 0)\), where a, b, and c are constants. When solving such equations, there are a few common methods:

• Factoring
• Completing the square
• Using the quadratic formula

During the solution process of \(\log(x+3) = \log\left(\frac{10}{x-2}\right)\), we eventually obtain the quadratic form \(x^2 + x - 16 = 0\).
Factoring is often the simplest method, provided that the equation is factorable over the integers. Here, it factors to \( (x - 4)(x + 4) = 0\), giving the roots \(x = 4\) and \(x = -4\).
This stage is crucial as it determines potential solutions to the problem, which then must be verified for validity given the context of logarithmic equations.

Once potential roots of an equation are found, especially in cases involving logarithms, it's vital to verify them for validity. This is because logarithms are only defined for positive values. Therefore, any solution leading to taking the logarithm of a non-positive number is invalid.
In this exercise, we found the roots to be \(x = 4\) and \(x = -4\). However, substituting \(x = -4\) into the original equation results in taking \(\log(-1)\), which is undefined.

• Always substitute the roots back into the original logarithmic expressions to check
• Ensure all conditions of the logarithms are met

Thus, the valid solution needs to fulfill all mathematical constraints present from the start of the problem. After checking, only \(x = 4\) remains as the valid solution. This step prevents incorrect answers from being accepted and ensures that the logic applied stays consistent with mathematical principles.