Problem 35 - Precalculus : Building Concepts and Connections

Question

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log x=0$$

Step-by-Step Solution

Answer

The solution to the equation \(\log x = 0\) is \(x = 1\).

Step 1: Understand the properties of logarithms

To start with, it is important to know that the logarithm of 1 to any base is always 0. That means \(\log_b 1 = 0\) for any base b.

Step 2: Compare with the given equation

The equation provided is \(\log x = 0\). From the property established in Step 1, we now know that a log of 1 to any base is 0, so the value of x should be 1.

Step 3: Determine any extraneous solutions

An extraneous solution is one that causes the argument of the logarithm to become negative or zero. But x=1 is a valid solution since the logarithm of 1 is defined, and equals zero. Hence, there are no extraneous solutions in this problem.

Key Concepts

Extraneous Solution

An extraneous solution is a solution that arises from the process of solving an equation which does not satisfy the original equation. When working with logarithmic equations, identifying and eliminating extraneous solutions is crucial to ensure the final answer is correct. Extraneous solutions often occur during the solving process, particularly when both sides of an equation are manipulated. It’s essential to verify potential solutions by substituting them back into the original equation. If any operation results in an argument of a logarithm being negative or zero, the corresponding solution is considered extraneous, as logarithms are only defined for positive numbers.

Properties of Logarithms

The properties of logarithms are essential tools for simplifying and solving logarithmic equations. The most fundamental property to remember is that the logarithm of 1 to any base is always 0, represented by the equation:

• \( \log_b 1 = 0 \)

Other properties include:

• The product rule: \( \log_b (MN) = \log_b M + \log_b N \)
• The quotient rule: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• The power rule: \( \log_b (M^n) = n \log_b M \)

These properties allow you to break down complex logarithmic expressions and solve them more easily. When working with equations like \( \log x = 0 \), knowing these basic properties helps to quickly identify that since \( \log_b 1 = 0 \), the value of \( x \) is simply 1.

Solving Equations

Solving a logarithmic equation involves several steps, guided by the properties of logarithms. Let’s break down the process:To solve \( \log x = 0 \):

• Understand that you need to find the value of \( x \) that makes this statement true.
• Based on the property of logarithms that \( \log_b 1 = 0 \) for any base \( b \), deduce that \( x = 1 \).

The solving process can include:

• Using properties of logarithms to simplify expressions.
• Isolating the logarithm if multiple functions are present.
• Checking for extraneous solutions by ensuring the argument of the log remains positive.

Always verify your solution by substituting it back into the original equation, ensuring every step aligns. If every condition is met, the solution is valid and correctly solves the equation.