Question

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

Short answer

Yes, it is possible for a logarithmic equation to have more than one extraneous solution. This can happen when the equation has multiple terms or multiple logarithmic functions, resulting in multiple solutions. If more than one of these solutions fall outside the domains of the involved logarithmic functions, they are considered extraneous solutions.

Step-by-step solution

Define the Domain of Logarithmic Functions

First, let's recall the domain of a logarithmic function: For any logarithmic function of the form \(y = \log_b x\) , the domain is \(x > 0\). This means logarithmic functions are only defined for positive values of \(x\). Therefore, if a solution does not respect this condition, it is considered extraneous.

Analyze Possibility of Multiple Extraneous Solutions

Consider a logarithmic equation with multiple terms or multiple logarithms - you can potentially end up with multiple solutions. Now, depending upon the restrictions of the domains of the involved logarithm functions, more than one of these solutions can fall outside these domains, rendering them as extraneous solutions.

Provide an Example

As an example, consider the equation \( \log_2(x-3) + \log_2(x+3) = 2 \). Solving this equation leads to the solutions \(x = 5\) and \(x = -1\). However, substituting \(x = -1\) back into the equation doesn't work out, as it would result in a negative argument for the logarithms, which is outside of their domain. Therefore, \(x = -1\) is an extraneous solution. By similar analysis, complex equations can yield more than one extraneous solutions.