Problem 46 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x.\) \(2 \log x=\log 2+\log (3 x-4)\)

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = 4\).

Step 1: Apply the Product Property

The product property of logarithms states that \( \log a + \log b = \log (a \times b) \). Apply this property to the right-hand side of the equation: \( \log 2 + \log (3x - 4) = \log (2(3x - 4)) \). Thus, the equation becomes \( 2 \log x = \log (2(3x - 4)) \).

Step 2: Simplify the Equation

Simplify the expression inside the logarithm \( \log (2(3x - 4)) = \log (6x - 8) \). Now the equation is \( 2 \log x = \log (6x - 8) \).

Step 3: Use the Power Rule

Apply the power rule, which states that \( a \log b = \log (b^a) \). Thus, \( 2 \log x = \log (x^2) \). Rewrite the equation as \( \log (x^2) = \log (6x - 8) \).

Step 4: Drop the Logarithms and Solve for x

If \( \log a = \log b \), then it follows that \( a = b \). This gives us the equation \( x^2 = 6x - 8 \). Rearrange it into a standard quadratic equation: \( x^2 - 6x + 8 = 0 \).

Step 5: Solve the Quadratic Equation

Factor the quadratic equation: \( x^2 - 6x + 8 = (x - 2)(x - 4) \). Set each factor equal to zero: \( x - 2 = 0 \) or \( x - 4 = 0 \). Solve these equations to find \( x = 2 \) or \( x = 4 \).

Step 6: Verify the Solutions

Check the solutions to ensure they satisfy the original equation. Substitute \( x = 2 \) into the equation: \( 2 \log 2 = \log 2 + \log (3(2) - 4) = \log 2 + \log 2 \), which is valid. Similarly, check \( x = 4 \): \( 2 \log 4 = \log 2 + \log (3(4) - 4) = \log 2 + \log 8 = \log 16 \), which is also valid. Therefore, both solutions work.

Key Concepts

Product Property of Logarithms

Logarithms have special properties that make them easier to work with, especially when dealing with multiplication and division inside the argument. One of these properties is the product property of logarithms. This property states:

• \( \log a + \log b = \log (a \times b) \)

In essence, if you have the sum of the logarithms of two numbers, you can combine them into a single logarithm by multiplying the numbers together inside the argument.
For instance, when we have the equation \( \log 2 + \log (3x - 4) \), it can be simplified using this property to \( \log (2(3x - 4)) \), which simplifies further once you multiply the terms. This step makes the equation easier to handle since it condenses the expression into fewer terms.

Power Rule of Logarithms

The power rule of logarithms is a fundamental tool for solving logarithmic equations. It is particularly useful when the logarithm is being multiplied by a number. The rule is stated as:

• \( a \log b = \log (b^a) \)

This rule allows us to move a multiplicative coefficient into the exponent of the argument inside the logarithm.
In the context of solving equations, like in the example \( 2 \log x \), it can be rewritten as \( \log (x^2) \). Utilizing this rule simplifies equations by transforming them into forms that are easier to compare or solve, especially when there's a need to cancel out the logarithm.

Quadratic Equations

Quadratic equations are a type of algebraic expression involving the variable raised to the second power. A standard quadratic equation is expressed in the form:

• \( ax^2 + bx + c = 0 \)

These equations are essential in algebra and can arise from a variety of contexts, such as the one discussed here where we equate \( x^2 = 6x - 8 \).
To convert this into a standard quadratic equation, you rearrange the terms, giving you:
\( x^2 - 6x + 8 = 0 \).
Understanding how to manage such equations is crucial for solving more complex mathematical problems.

Factoring Quadratics

Factoring is a method of solving quadratic equations by expressing the quadratic expression as a product of two binomials. For example, the quadratic equation \( x^2 - 6x + 8 = 0 \) can be factored as follows:

• Look for two numbers that multiply to \(c\) (constant term) and add up to \(b\) (coefficient of \(x\)). For \(x^2 - 6x + 8\), the numbers are \(-2\) and \(-4\).
• Rewrite the quadratic as \((x - 2)(x - 4) = 0\).

Once the quadratic is factored, each factor can be set to zero to solve for \(x\):

• \(x - 2 = 0 \rightarrow x = 2\)
• \(x - 4 = 0 \rightarrow x = 4\)

Factoring is not only a method of finding the roots of quadratic equations but also provides insight into the graph of the function by showing its x-intercepts. Always verify that the solutions work in the original equation to ensure validity.