Order The Steps To Solve The Equationlog(x2 - 15) = Log(2x) Form 1 To 5.x2 - 2x - 15 = 0Potential Solutions

Order The Steps To Solve The Equationlog(x2 - 15) = Log(2x) Form 1 To 5.x2 - 2x - 15 = 0Potential Solutions

Understanding how to solve logarithmic equations is a fundamental skill in algebra, especially when dealing with equations involving multiple logarithmic expressions. The equation given, log(x² - 15) = log(2x), is a classic example where logarithmic properties can be applied to simplify and solve for the variable x. This article provides a comprehensive, step-by-step guide to solving this specific logarithmic equation, emphasizing the importance of proper solution validation, domain considerations, and potential solutions. Whether you are a student preparing for exams or a mathematics enthusiast looking to deepen your understanding, mastering this process will enhance your problem-solving toolkit.

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Introduction to Logarithmic Equations

Logarithmic equations are equations where the variable appears inside a logarithm. These equations often arise in real-world contexts such as exponential growth, decay problems, and in various scientific applications. Solving these equations requires understanding the properties of logarithms and the domain restrictions that come with them.

The basic properties of logarithms include:

• Product Rule: logb(xy) = logb(x) + log_b(y)


• Quotient Rule: logb(x/y) = logb(x) - log_b(y)


• Power Rule: logb(x^k) = k logb(x)


• Change of Base: logb(x) = logk(x) / log_k(b)

In algebra, the most common base for logarithms is 10 (common logarithm) or e (natural logarithm), but the principles are applicable across bases as long as the base is positive and not equal to 1.

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Understanding the Given Equation

The specific equation under consideration is:

log(x² - 15) = log(2x)

This equation involves two logarithmic expressions set equal to each other. To solve for x, we will use properties of logarithms and algebraic techniques.

Key considerations:

• The logs are equal when their arguments are equal, provided those arguments are within the domain.


• The arguments of the logarithms must be positive because the logarithm function is only defined for positive real numbers.

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Step-by-Step Solution Process

Step 1: Understand the Domain Restrictions

Before manipulating the equation, identify the domain restrictions to ensure the solutions are valid.

• For log(x² - 15): x² - 15 > 0 → x² > 15 → x > √15 or x < -√15


• For log(2x): 2x > 0 → x > 0

Combined domain restrictions:

• x > 0 (from log(2x))


• x > √15 (~3.872) or x < -√15 (from log(x² - 15))

Since x must satisfy both, the feasible domain is:

x > √15 (approximately x > 3.872)

The negative values are invalid because 2x would be negative or zero, making log(2x) undefined.

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Step 2: Set the Logarithmic Arguments Equal

Since log functions are equal and their bases are the same, their arguments must be equal:

x² - 15 = 2x

This simplifies the equation from logarithmic form to algebraic form.

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Step 3: Rearrange the Equation

Bring all terms to one side to set the quadratic to zero:

x² - 2x - 15 = 0

This is a quadratic equation in standard form.

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Step 4: Solve the Quadratic Equation

Use the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where a = 1, b = -2, c = -15.

Calculate the discriminant:

D = b² - 4ac = (-2)² - 4 1 (-15) = 4 + 60 = 64

Calculate the roots:

x = [2 ± √64] / 2

x = [2 ± 8] / 2

• For the positive root:

x = (2 + 8) / 2 = 10 / 2 = 5

• For the negative root:

x = (2 - 8) / 2 = (-6) / 2 = -3

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Step 5: Validate Solutions Against Domain Restrictions

Recall the domain restrictions: x > √15 (~3.872)

• x = 5: satisfies x > 3.872 → Valid


• x = -3: does not satisfy x > 3.872, and also, 2 (-3) = -6, which makes log(2x) undefined → Invalid

Conclusion: The only potential solution is x = 5.

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Potential Solutions and Final Check

After identifying x = 5 as the candidate solution, verify it by substituting back into the original equation:

log( (5)² - 15 ) = log(2 5)

Calculate each side:

• Left side: log(25 - 15) = log(10)


• Right side: log(10)

Since both sides are equal, x = 5 satisfies the original equation.

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Summary of the Solution Steps

1. Identify domain restrictions:

• x > √15
• x > 0

2. Set the logarithmic arguments equal:

• x² - 15 = 2x

3. Rearranged to quadratic form:

• x² - 2x - 15 = 0

4. Solve the quadratic:

• Roots: x = 5 and x = -3

5. Validate solutions against domain restrictions:

• Only x = 5 is valid

6. Verify the solution by substitution:

• Confirmed x = 5 satisfies the original equation

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Additional Tips for Solving Logarithmic Equations

• Always check the domain restrictions after solving algebraic equations involving logarithms.
• Remember that the logarithm of a negative number or zero is undefined.
• When setting the arguments equal, be cautious of extraneous solutions that might not satisfy the original equation.
• Use the quadratic formula for quadratic equations, but consider factoring if possible to simplify calculations.

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Conclusion

Solving the logarithmic equation log(x² - 15) = log(2x) involves a systematic approach: understanding the domain restrictions, transforming the equation into a quadratic form, solving for potential solutions, and validating these solutions against the original constraints. The only valid solution in this case is x = 5, which satisfies all conditions and makes the original logarithmic expressions valid.

Mastering these steps enhances your ability to tackle similar logarithmic equations efficiently. Remember, always verify your solutions to avoid extraneous roots, especially when dealing with the properties of logarithms and their domains. With practice, solving such equations becomes an intuitive process, empowering you to handle more complex algebraic and logarithmic problems confidently.