Problem 65 - A Graphical Approach to College Algebra

Question

A student told a friend, "You must reject any negative solution of an equation involving logarithms." Is this correct? Write an explanation of your answer.

Step-by-Step Solution

Answer

Reject solutions causing negative arguments in logarithms.

Step 1: Understanding the Statement

The student suggests that any negative solution in equations involving logarithms should be rejected. We need to evaluate if this is always correct and under what conditions it applies.

Step 2: Review Properties of Logarithms

Recall that the domain of a logarithmic function, such as \( \log_b(x) \), requires \( x > 0 \) and \( b > 0 \), \( b eq 1 \). Logarithms are undefined for non-positive arguments.

Step 3: Analyze the Argument of the Logarithm

When solving equations involving logarithms, ensure that the argument of the logarithmic function remains positive. Example: if \( \log(x) = y \), then \( x > 0 \).

Step 4: Evaluate Solutions

Solutions to logarithmic equations may not necessarily be negative themselves. However, if substituting a found solution results in a negative argument inside a logarithm, that solution is extraneous and must be rejected.

Step 5: Consider Specific Examples

Consider an equation like \( \log(x-2) = 1 \). Solve to find \( x - 2 = 10 \rightarrow x = 12 \). This solution is valid as \( 12 - 2 = 10 \), which is positive. If a calculation resulted in a negative argument, it would be invalid.

Step 6: Conclusion

Negative solutions are not automatically dismissed; only those that result in a negative argument within the logarithm are invalid. It depends on the context and the specific equation.

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is essential when dealing with logarithmic equations. A logarithm, represented as \( \log_b(x) \), can be thought of as the inverse of exponentiation. This means if \( b^y = x \), then \( \log_b(x) = y \). For this idea to work consistently, we need some rules:

• The base \( b \) must be greater than zero and not equal to one \( (b > 0, \, b eq 1) \).
• The argument \( x \) must be positive \( (x > 0) \).

When solving logarithmic equations, it's crucial to keep these properties in mind to avoid errors. If the base or the argument doesn't meet these conditions, the logarithm becomes undefined, and the equation invalid. Always ensure that base and argument satisfy these criteria when manipulating or solving logarithmic expressions.

Logarithmic Function Domain

The domain of a logarithmic function is the set of all input values \( x \) that can produce a valid output y. For the logarithmic function \( \log_b(x) \), the domain consists entirely of positive real numbers. This is because a logarithm is undefined for zero or negative numbers.

• The mathematical representation of the domain is \( x > 0 \).
• This constraint ensures the safe operation of the logarithmic function.

When solving logarithmic equations, it's vital to keep the domain in mind. Sometimes a solution will appear valid but will produce an argument \( x \) that is outside the domain of possible values for the logarithmic function. Such solutions cannot be used as they will make the logarithm undefined,

Extraneous Solutions

Extraneous solutions are those solutions that arise from the algebraic manipulation of an equation but do not satisfy the original condition set by the equation. When solving logarithmic equations, it's common to encounter extraneous solutions due to the transformations applied to solve them.A solution to a logarithmic equation is considered extraneous if:

• It results in a zero or negative value when substituted back into the argument of the logarithmic expression.

For example, if you solve an equation and find \( x = -3 \) or \( x = 0 \), substituting these values back into the logarithmic component \( \log(x) \) would result in something undefined. These solutions, however mathematically obtained, are not valid for the original equation.To avoid mistakes, always substitute potential solutions back into the original equation to ensure they are valid and within the permissible domain. Only keep those solutions that maintain a correct argument for the logarithmic function.