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0 5 Log X-1(x^2-8x+16)

4 min read Jun 17, 2024
0 5 Log X-1(x^2-8x+16)

Solving the Logarithmic Equation: 0.5 log<sub>x-1</sub>(x<sup>2</sup>-8x+16)

This article will guide you through solving the logarithmic equation 0.5 log<sub>x-1</sub>(x<sup>2</sup>-8x+16). We will break down the steps and explain the concepts involved.

Understanding the Equation

First, let's analyze the given equation:

  • Logarithmic Function: The equation involves a logarithmic function with base (x-1).
  • Argument: The argument of the logarithm is (x<sup>2</sup>-8x+16), which is a quadratic expression.

Solving the Equation

  1. Simplify the Equation:

    • Rewrite 0.5 as 1/2.
    • The equation now becomes: (1/2) log<sub>x-1</sub>(x<sup>2</sup>-8x+16) = 0
    • Multiply both sides by 2: log<sub>x-1</sub>(x<sup>2</sup>-8x+16) = 0

  2. Convert to Exponential Form:

    • Recall that log<sub>b</sub>a = c is equivalent to b<sup>c</sup> = a.
    • Applying this to our equation, we get: (x-1)<sup>0</sup> = x<sup>2</sup>-8x+16

  3. Solve the Equation:

    • Any number raised to the power of 0 equals 1. So, (x-1)<sup>0</sup> = 1
    • Our equation now becomes: 1 = x<sup>2</sup>-8x+16
    • Subtract 1 from both sides: 0 = x<sup>2</sup>-8x+15
    • Factor the quadratic expression: 0 = (x-3)(x-5)
    • Therefore, x = 3 or x = 5.

  4. Check for Extraneous Solutions:

    • Important: We must check if our solutions satisfy the original equation's domain.
    • Domain Restrictions: For a logarithmic function, the argument (x<sup>2</sup>-8x+16) must be greater than 0, and the base (x-1) must be greater than 0 and not equal to 1.
    • Checking x = 3:
      • Base: (3-1) = 2 (valid)
      • Argument: (3<sup>2</sup>-8(3)+16) = 1 (valid)
    • Checking x = 5:
      • Base: (5-1) = 4 (valid)
      • Argument: (5<sup>2</sup>-8(5)+16) = 1 (valid)

  5. Final Solution:

    • Both solutions, x = 3 and x = 5, satisfy the domain restrictions of the original equation.
    • Therefore, the solutions to the equation 0.5 log<sub>x-1</sub>(x<sup>2</sup>-8x+16) are x = 3 and x = 5.

Conclusion

We successfully solved the logarithmic equation by simplifying, converting to exponential form, and carefully considering the domain restrictions. Remember to always check your solutions against the domain of the original equation to avoid extraneous solutions.

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