Problem 97 - Algebra and Trigonometry

Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)} $$

Step-by-Step Solution

Answer

The given equation is false. The correct equation should be \(\log (x+3) - \log (2x) = \log \left(\frac{x+3}{2x}\right)\)

Step 1: Analyze the Equation

First, observe that the logarithmic equation \(\log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)}\) includes subtraction on the left side. In logarithms, when two logs with the same base are subtracted, their arguments get divided. So you can rewrite the left side of the equation as \(\log \left(\frac{x+3}{2x}\right)\).

Step 2: Compare both sides of the Equation

Now, compare the revised equation \(\log \left(\frac{x+3}{2x}\right)\) to the right side, which is \(\frac{\log (x+3)}{\log (2x)}\). These are different expressions; they are not equivalent for all values of x. Therefore, the original statement is false.

Step 3: Correct the Equation

Since the statement is found to be false, you need to make the correct adjustment to make it true. Using the logarithm subtraction rule mentioned in Step 1, the correct equation should be \(\log (x+3) - \log (2x) = \log \left(\frac{x+3}{2x}\right)\).

Key Concepts

Logarithm Properties

Logarithm properties are essential tools that make it easier to solve logarithmic equations and understand their behavior. One key property is that the subtraction of two logarithms with the same base can be expressed as the logarithm of a quotient. Specifically, given two logarithms, \(\log_b (A) - \log_b (B)\), they can be combined into the single logarithm \(\log_b\left(\frac{A}{B}\right)\). This is known as the quotient property of logarithms, and it helps simplify expressions involving logarithms.

Another important property is the power rule, which states that the logarithm of a power can be rewritten by bringing the exponent in front of the log. In mathematical terms, \(\log_b(A^n) = n \cdot \log_b(A)\). There is also the product property, where \(\log_b (A) + \log_b (B) = \log_b(AB)\). These rules are not just theoretical but serve practical purposes in making complex logarithmic equations more manageable.

Being comfortable with these properties allows you to apply them correctly and confidently to different questions and equations involving logarithms.

Equation Solving

Equation solving often involves manipulating an equation into a form that reveals its solution. For logarithmic equations, the key is using logarithm properties to simplify the expression. In the given example, we started with the expression \( \log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)} \). The left side of this equation can be rephrased into a single logarithm using the quotient property: \( \log \left(\frac{x+3}{2x}\right) \).

Once simplified, it becomes clearer that the left and right sides of the equation were not equivalent. Solving logarithmic equations often involves simplifying complex expressions into forms that reveal whether both sides are identical or not. This requires both the understanding of logarithms and the ability to manipulate algebraic expressions.

• Identify the properties of logarithms that can be applied to the equation.
• Simplify each side of the equation as much as possible.
• Check for equivalence by comparing simplified expressions.

Understanding these steps can make the process of solving such equations smoother and more intuitive.

Mathematical Proofs

Mathematical proofs are a logical step-by-step approach used to verify the truth of a statement or proposition. In the context of logarithmic equations, it involves demonstrating whether an equation holds true under specific conditions. For example, when presented with the false statement \( \log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)} \), we proved it false by simplifying both sides and finding they were not the same.

Proofs often involve:

• Clearly stating the proposition or equation.
• Using known properties and theorems to manipulate the expression.
• Arriving at a contradiction or confirming equivalence.
• Concluding based on the findings.

In the case here, we compared the simplified left side, \( \log \left(\frac{x+3}{2x}\right) \), with the given right side, and since they were unequal, we concluded the statement was false. Understanding proofs ensures that you approach problems with logical reasoning instead of mere guesswork, reinforcing the robustness of mathematics.