Problem 348 - Precalculus

Question

Solve the equation for \(x,\) if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log (4-2 x)=\log (-4 x)\)

Step-by-Step Solution

Answer

The solution is \( x = -2 \); graphs intersect at this point.

Step 1: Understand the Equation

The equation given is \( \log (4 - 2x) = \log (-4x) \). Both sides of the equation are logarithmic, which suggests that we may be able to solve this equation by equating the arguments of the logarithms.

Step 2: Use the Property of Logarithms

Since \( \log(a) = \log(b) \) implies that \( a = b \), we can set the arguments equal to each other: \( 4 - 2x = -4x \).

Step 3: Rearrange the Equation

To solve \( 4 - 2x = -4x \) for \( x \), start by moving all terms involving \( x \) to one side and constants to the other: \( 4 = -4x + 2x \).

Step 4: Simplify the Equation

Simplify the expression by combining like terms: \( 4 = -2x \).

Step 5: Solve for \(x\)

Divide both sides by \(-2\) to solve for \( x \): \( x = -2 \).

Step 6: Check the Solution for Validity

Verify if \( x = -2 \) results in valid logarithmic expressions. For \( \log(4 - 2(-2)) = \log(8) \) and \( \log(-4(-2)) = \log(8) \). Both are valid since \( \log(8) \) is defined.

Step 7: Graph the Functions

Graph the functions \( y = \log(4 - 2x) \) and \( y = \log(-4x) \) on the same coordinate plane. The graph of both functions should intersect at \( x = -2 \).

Step 8: Verify the Point of Intersection

Observe the graph to ensure that the point \( (x = -2, y = \log(8)) \) is a point of intersection, confirming the solution is correct.

Key Concepts

Graphical Verification

Graphs can be a powerful tool to visually verify solutions to equations. When you graph functions related to equations, the points where they intersect represent solutions. For the given problem, you are asked to graph two logarithmic functions:

• \( y = \log(4 - 2x) \)
• \( y = \log(-4x) \)

Once graphed, the intersection point directly shows where both expressions equal each other in terms of their corresponding x-values. In this case, both functions should intersect at the point \( x = -2 \) where the y-value is \( \log(8) \).
When you look at the graph, seeing this intersection confirms that \( x = -2 \) is indeed a valid solution. This visual confirmation adds a layer of reliability to the algebraic solution, ensuring that you haven’t made calculation mistakes or overlooked the domain restrictions of the logarithmic functions.

Properties of Logarithms

Logarithms have distinct properties that can help solve complex equations by simplifying expressions. One key property used in solving the equation \( \log(a) = \log(b) \) is that it implies \( a = b \). This property is central and comes from the definition of logarithms as exponents.
You leveraged this property in the exercise to equate the arguments of the logarithms directly:

• From \( \log(4 - 2x) = \log(-4x)\), you set \( 4 - 2x = -4x \).

Besides this vital property, there are other useful properties such as:

• \( \log(ab) = \log(a) + \log(b) \)
• \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)
• \( \log(a^b) = b \log(a) \)

Understanding these allows you to manipulate and simplify logarithmic expressions, making solving equations more manageable.
These rules are not only instrumental in mathematical calculations but are essential for tackling scientific problems involving exponential growth or decay.

Solving Equations

Solving logarithmic equations involves using a systematic approach to isolate the variable, similar to solving any other algebraic equation. When faced with \( \log(4 - 2x) = \log(-4x) \), you first use properties of logarithms to equate the arguments directly.
Here's a step-by-step approach to solving the equation:

• Start by applying the property that says \( \log(a) = \log(b) \) implies \( a = b \).
• Simplify and rearrange the equation to get all terms involving x on one side: \( 4 - 2x = -4x \) can be rearranged to \( 4 = -2x \).
• Next, solve for \( x \) by isolating it: divide both sides by \(-2\) to find \( x = -2 \).

Remember, it’s also important to verify your solution to ensure it is within the domain of the original logarithmic expressions. For example, check if substituting back gives a defined value of a logarithm.
This precise and careful process of solving logarithmic equations guarantees correctness and deepens understanding of the underlying mathematical principles.