91Ó°ÊÓ

Understanding the properties of logarithms is crucial when solving logarithmic equations. A logarithm is the power to which a number (the base) must be raised to obtain another number. For example, if we have the equation \(\log_b(x) = y\), this can be interpreted as \(b^y = x\). When solving equations involving logarithms, there are several key properties we use:

• Product Property: The log of a product is the sum of the logs of the factors, which we use in Step 1 of the exercise. Mathematically, \(\log_b(mn) = \log_b(m) + \log_b(n)\).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms. That is, \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\).
• Power Property: \(\log_b(m^n) = n\log_b(m)\), which tells us that a logarithm of a power allows the exponent to be brought in front as a multiplier.

Recognizing and applying these properties properly is essential for simplifying and solving logarithmic equations efficiently.

Transforming a logarithmic equation into its exponential form simplifies the process of finding the variable. This conversion is based on the definition of a logarithm. The equation \(\log_b(x) = y\) can be rewritten as \(b^y = x\), where \(b\) is the base of the logarithm, \(y\) is the logarithm of the number \(x\), and \(x\) is the result when \(b\) is raised to the power of \(y\).

In Step 2 of our exercise, we use this principle to write \(\log [(x+1)(x-1)] = 0\) as \(1^0 = (x+1)(x-1)\). Since any number raised to the power of zero is 1, it simplifies the equation to a form where we can easily solve for \(x\) by expanding and rearranging terms.

An extraneous solution is a result obtained during the process of solving an equation that is not a valid solution to the original equation. In logarithmic equations, extraneous solutions often arise when the logarithm of a negative number or zero occurs, which is undefined, as logarithms are only defined for positive numbers.

In Step 4 of our solution, we must verify the potential solutions \(x = ±1\) in the original equation. The term \(\log (x-1)\) becomes undefined when \(x = -1\), as the argument of the logarithm must be positive. Thus, \(x = -1\) is an extraneous solution and is discarded, leaving us with the valid solution \(x = 1\). It is always important to check for extraneous solutions to ensure that all solutions satisfy the original equation.