Equations in which the power has to be found

Equations in which the power has to be found

Introduction to Solving Equations with Unknown Exponents

  • Mathematics occasionally presents problems where the variable to be solved for resides in an exponent.
  • Solving for powers can be frequently encountered in exponential growth or decay models, compound interest calculations, or when dealing with logarithmic scales.
  • The key to solving such problems is to recognize the common bases in the equation and then equate the exponents.

Rules for Equations With Unknown Exponents

  • If two exponential expressions with the same base are equal, then their exponents must be equal. This rule is crucial when we want to solve equations involving powers.

Steps to Solve Power Equations

  1. Express each side of the equation with the same base.
  2. Once the bases are the same, set the exponents of the equation equal to each other.
  3. Then essentially the question has been transformed into a simple algebraic equation. Use algebra to solve for the unknown variable.

Example Problem

  • Consider the equation 2^x = 8.
  • First, try to express 8 as 2 raised to some power. In this case, 2^3 = 8.
  • This allows us to rewrite the equation as 2^x = 2^3.
  • Now that the bases are equal, we can assert that x must be 3.
  • The solution to the equation is therefore x = 3.

Final Notes

  • Make sure to check your answers by substiting the found value back into the initial equation.
  • If necessary, logarithms can be helpful for solving equations with unknown exponents, but understanding and comfort with logarithms is typically introduced later at A-level maths.
  • Always remember to express equation to the same base as it is important when dealing with equations involving powers.