Solve equations with variable exponents

Key notes:

Understanding Exponents

• Exponents represent repeated multiplication (e.g., 2³ = 2 × 2 × 2 = 8 ).
• The base is the number being multiplied, and the exponent tells how many times to multiply it.

Equations with Variable Exponents

• These are equations where the exponent contains a variable, such as xⁿ = a or 2^x = 16.
• The goal is to find the value of the variable that makes the equation true.

Solving Equations with the Same Base

If the equation has the same base on both sides, set the exponents equal to each other.

• Example: 3^x = 3⁴ ⟹ x = 4.

Using Logarithms (Basic Introduction)

If the equation is not easily rewritten with the same base, logarithms can be used to solve for the exponent.

• Example: 2^x =10 ⟹ x = log⁡₂ 10 (Grade 7 may use trial and error for estimating x).

Exponent Rules for Solving Equations

• Multiplication Rule: a^m × aⁿ = a^m+n
• Division Rule: a^m / aⁿ = a^m−n.
• Power of a Power Rule: (a^m)ⁿ = a^m×n.
• Zero Exponent Rule: a⁰ = 1 (where a ≠ 0).

Checking Solutions

• Substitute the found value of the variable back into the original equation to verify the solution.

Applications in Real Life

• Growth and decay problems (e.g., population growth, radioactive decay).
• Financial calculations involving compound interest.