What Is The Solution Of Log6x Log6 X 5 2

The equation log₆x log₆x = 5 + 2 represents a mathematical problem that requires a solid understanding of logarithms and algebraic manipulation to solve. Understanding the fundamental properties of logarithms and applying the correct algebraic techniques are key to finding the value of x that satisfies this equation.

Understanding the Logarithmic Equation

The given equation, log₆x log₆x = 5 + 2, involves logarithms with base 6. The logarithm log₆x represents the power to which 6 must be raised to obtain x. To solve this equation, we need to simplify it and isolate x using the properties of logarithms.

Steps to Solve the Logarithmic Equation

Here's a detailed breakdown of the steps to solve the equation:

1. Simplify the Equation

   The first step is to simplify both sides of the equation:

   log₆x log₆x = 5 + 2

   This can be written as:

   (log₆x)² = 7

2. Take the Square Root of Both Sides

   To remove the square from the left side, take the square root of both sides of the equation:

   √((log₆x)²) = ±√7

   log₆x = ±√7

   This gives us two possible equations:

   • log₆x = √7
   • log₆x = -√7

3. Convert Logarithmic Form to Exponential Form

   To solve for x, convert the logarithmic equations to their equivalent exponential forms. The general form of a logarithm logₐb = c is equivalent to aᶜ = b. Applying this to our equations:

   • For log₆x = √7: x = 6^(√7)
   • For log₆x = -√7: x = 6^(-√7)

4. Calculate the Values of x

   Now, calculate the values of x using a calculator:

   • x = 6^(√7) ≈ 6^(2.64575) ≈ 83.325
   • x = 6^(-√7) ≈ 6^(-2.64575) ≈ 0.012

   Thus, we have two possible solutions for x:

   • x ≈ 83.325
   • x ≈ 0.012

5. Verify the Solutions

   It is crucial to verify the solutions by substituting them back into the original equation to ensure they are valid.

   • For x ≈ 83.325:
     • log₆(83.325) ≈ 2.64575
     • (2.64575)² ≈ 7
     • The solution is valid.
   • For x ≈ 0.012:
     • log₆(0.012) ≈ -2.64575
     • (-2.64575)² ≈ 7
     • The solution is valid.

   Both solutions satisfy the original equation.

Advanced Insights into Solving Logarithmic Equations

Understanding Logarithmic Properties

Logarithmic properties are fundamental to solving logarithmic equations. Here's a review of some essential properties:

• Product Rule: logₐ(mn) = logₐm + logₐn
• Quotient Rule: logₐ(m/n) = logₐm - logₐn
• Power Rule: logₐ(m^k) = k logₐm
• Change of Base Formula: logₐb = log꜀b / log꜀a
• Inverse Property: a^(logₐx) = x and logₐ(a^x) = x

Common Mistakes to Avoid

When solving logarithmic equations, it's easy to make mistakes. Here are some common pitfalls to watch out for:

• Incorrectly Applying Logarithmic Properties: Ensure you apply the properties correctly. For instance, the product rule is often confused with the logarithm of a sum.
• Forgetting to Check Solutions: Always verify your solutions in the original equation, especially because logarithmic functions have domain restrictions. You cannot take the logarithm of a negative number or zero.
• Ignoring the Domain of Logarithms: Be mindful that the argument of a logarithm must be positive. If a solution results in taking the logarithm of a non-positive number, it is an extraneous solution and must be discarded.
• Algebraic Errors: Double-check your algebraic manipulations to avoid mistakes in simplification or solving for x.

Practical Examples and Applications

Logarithmic equations are not just theoretical exercises; they have numerous practical applications in various fields:

• Finance: Compound interest calculations often involve logarithmic equations. For example, determining the time it takes for an investment to double at a certain interest rate.
• Science: Radioactive decay and pH calculations in chemistry use logarithmic scales. The decay of a radioactive substance can be modeled using exponential functions, which are often solved using logarithms.
• Engineering: Signal processing and acoustics use logarithmic scales to measure sound intensity and signal strength.
• Computer Science: Analysis of algorithms often involves logarithmic functions, particularly in algorithms that use divide-and-conquer strategies like binary search.

The Significance of Base in Logarithmic Equations

The base of a logarithm is crucial. Different bases can significantly affect the solution of an equation. In the given equation, the base is 6, but let's consider how different bases would influence the equation and its solutions.

• Base 10 (Common Logarithm): If the base were 10, the equation would be log₁₀x log₁₀x = 7, and the solutions would be x = 10^(√7) and x = 10^(-√7).
• Base e (Natural Logarithm): If the base were e, the equation would be ln(x) ln(x) = 7, and the solutions would be x = e^(√7) and x = e^(-√7).

The base affects the exponential form and the final value of x. Understanding the impact of the base is essential for accurate problem-solving.

Advanced Techniques for Solving Logarithmic Equations

Using Substitution

In more complex logarithmic equations, substitution can simplify the problem. For instance, consider the equation:

log₂(x)² - 3log₂(x) + 2 = 0

Here, let y = log₂(x). The equation becomes:

y² - 3y + 2 = 0

This is a quadratic equation that can be easily solved for y:

(y - 1)(y - 2) = 0

So, y = 1 or y = 2. Now, substitute back to find x:

• log₂(x) = 1 => x = 2¹ = 2
• log₂(x) = 2 => x = 2² = 4

Thus, the solutions are x = 2 and x = 4.

Dealing with Extraneous Solutions

Extraneous solutions are solutions that satisfy the transformed equation but not the original equation. They often arise when squaring both sides of an equation or applying logarithmic properties that change the domain. Always check your solutions in the original equation.

Consider the equation:

log₂(x - 1) + log₂(x + 1) = 3

Using the product rule:

log₂((x - 1)(x + 1)) = 3

log₂(x² - 1) = 3

Convert to exponential form:

x² - 1 = 2³

x² - 1 = 8

x² = 9

x = ±3

Now, check the solutions:

• For x = 3:
  • log₂(3 - 1) + log₂(3 + 1) = log₂(2) + log₂(4) = 1 + 2 = 3 (Valid)
• For x = -3:
  • log₂(-3 - 1) + log₂(-3 + 1) = log₂(-4) + log₂(-2) (Not Valid)

Since we cannot take the logarithm of a negative number, x = -3 is an extraneous solution. The only valid solution is x = 3.

Graphical Solutions

Graphical methods can also be used to solve logarithmic equations, especially when analytical solutions are difficult to find. By plotting the functions on both sides of the equation and finding their intersection points, we can approximate the solutions.

For the original equation (log₆x)² = 7, we can plot y = (log₆x)² and y = 7 on the same graph. The x-coordinates of the intersection points will give the approximate solutions.

The Underlying Mathematical Principles

The solution of logarithmic equations is deeply rooted in the fundamental principles of mathematics.

Exponential Functions

Logarithms are the inverse of exponential functions. An exponential function has the form f(x) = aˣ, where a is the base. The logarithm answers the question: "To what power must a be raised to obtain x?" This inverse relationship is critical in solving logarithmic equations.

Algebraic Manipulation

Algebraic manipulation is essential for isolating variables and simplifying equations. Techniques such as taking square roots, factoring, and applying the distributive property are used extensively.

Function Domains and Ranges

Understanding the domains and ranges of functions is crucial. The domain of a logarithmic function is all positive real numbers, while the range is all real numbers. This knowledge helps in identifying extraneous solutions.

Common Questions About Logarithmic Equations (FAQ)

• What is a logarithm?

  A logarithm is the inverse operation to exponentiation. It answers the question of what exponent is needed to raise a base to a certain number.

• How do I solve logarithmic equations?

  To solve logarithmic equations:

  1. Simplify the equation using logarithmic properties.
  2. Convert the logarithmic equation to exponential form.
  3. Solve for the variable.
  4. Check your solutions in the original equation.

• What are extraneous solutions?

  Extraneous solutions are solutions that satisfy a transformed equation but not the original equation. They often arise due to domain restrictions or algebraic manipulations.

• Can I take the logarithm of a negative number?

  No, you cannot take the logarithm of a negative number or zero because the domain of a logarithmic function is all positive real numbers.

• Why is it important to check solutions in logarithmic equations?

  Checking solutions is important to identify and discard extraneous solutions that do not satisfy the original equation due to domain restrictions or algebraic manipulations.

• How does the base of a logarithm affect the solution?

  The base significantly affects the solution. Different bases will result in different exponential forms and final values for the variable.

• What are some common mistakes to avoid when solving logarithmic equations?

  Common mistakes include incorrectly applying logarithmic properties, forgetting to check solutions, ignoring the domain of logarithms, and making algebraic errors.

Conclusion

Solving the logarithmic equation log₆x log₆x = 5 + 2 involves simplifying the equation, taking the square root of both sides, converting the equation to exponential form, and verifying the solutions. The solutions are x ≈ 83.325 and x ≈ 0.012. Understanding the properties of logarithms, avoiding common mistakes, and recognizing the importance of the base are crucial for successful problem-solving. Logarithmic equations have broad applications in finance, science, engineering, and computer science, making their understanding essential for various fields. By mastering the techniques and principles discussed, you can confidently tackle a wide range of logarithmic equations.