(1/16)^x+3=(1/4)^x+1

Jasonbradley

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Jun 16, 2024

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Solving the Exponential Equation: (1/16)^x+3 = (1/4)^x+1

This article will guide you through solving the exponential equation (1/16)^x+3 = (1/4)^x+1. We'll use the properties of exponents to simplify the equation and then solve for the unknown variable, x.

Understanding the Properties of Exponents

Before we begin, let's review some key properties of exponents:

• Product of Powers: a^m * a^n = a^(m+n)
• Quotient of Powers: a^m / a^n = a^(m-n)
• Power of a Power: (a^m)^n = a^(m*n)

Solving the Equation

1. Express both sides with the same base: Notice that both (1/16) and (1/4) can be expressed as powers of 4:

   • (1/16) = (1/4)^2
   • (1/4) = (1/4)^1

   Substituting these into the original equation gives us: (1/4)^(2(x+3)) = (1/4)^(x+1)

2. Equate the exponents: Since the bases are now the same, we can equate the exponents: 2(x+3) = x + 1

3. Solve for x: 2x + 6 = x + 1 2x - x = 1 - 6 x = -5

Conclusion

Therefore, the solution to the exponential equation (1/16)^x+3 = (1/4)^x+1 is x = -5.