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Solving Exponent Equations: 9 to the 4x Equals One Third

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Solving Exponent Equations: 9 to the 4x Equals One Third

In the realm of algebra, exponent equations present a unique challenge, often requiring a blend of understanding and strategic manipulation. Today, we will delve into a specific problem: solving for x in the equation 94x = 1/3. This equation might seem daunting at first, but with the right approach, it becomes a straightforward exercise in applying the properties of exponents and logarithms.

Understanding the Problem

The equation 94x = 1/3 essentially asks: "To what power must we raise 9 to obtain the value of one-third?" To solve this, we need to express both sides of the equation in a way that allows us to isolate x.

Steps to Solve

  1. Express 9 and 1/3 as powers of the same base: Both 9 and 1/3 can be expressed as powers of 3. We know that 9 is equal to 32 and 1/3 is equal to 3-1.
  2. Substitute the base values: Substituting these values into the original equation, we get: (32)4x = 3-1.
  3. Simplify using exponent rules: Remember that when raising a power to another power, we multiply the exponents. Therefore, the equation becomes: 38x = 3-1.
  4. Equate the exponents: Since the bases are now the same, we can equate the exponents: 8x = -1.
  5. Solve for x: Dividing both sides by 8, we get: x = -1/8.

Solution

Therefore, the solution to the equation 94x = 1/3 is x = -1/8. This means that 9 raised to the power of 4 times -1/8 (which is -1/2) equals 1/3.

Key Concepts

  • Exponents: An exponent indicates how many times a base number is multiplied by itself.
  • Logarithms: Logarithms are the inverse of exponents. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number.
  • Properties of Exponents: Understanding the properties of exponents is crucial for solving exponent equations. For example, (am)n = am*n.

Conclusion

Solving exponent equations like 94x = 1/3 involves understanding the properties of exponents and logarithms. By expressing the numbers as powers of the same base and applying the appropriate exponent rules, we can effectively isolate the variable and determine its value. Mastering these concepts empowers you to navigate the challenges of algebra and unlock a deeper understanding of mathematical relationships.

Logarithms