Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Do not use a calculator. $$12^{x-3}=1$$

Step-by-Step Solution

Answer

The solution is \(x=3\).

Step 1: Identify Properties of Exponents

The equation we are given is \(12^{x-3}=1\). Recall that any number raised to the power of zero equals 1. Therefore, the equation \(12^0 = 1\) tells us that the exponent must be zero: \(x-3=0\).

Step 2: Solve for the Variable

Now that we have established \(x-3=0\), solve for \(x\) by adding 3 to both sides of the equation. This gives us \(x=3\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication. When you see a term like \(12^{x-3}\), it means that 12 is multiplied by itself \(x-3\) times. Understanding exponent rules is crucial for solving equations involving them. One key rule is that any non-zero number raised to the zero power is 1: \(a^0 = 1\). This is why \(12^{x-3}\) equaling 1 suggests that \(x-3\) must be zero. It's like a little puzzle where you know the outcome and deduce what the exponent must be to make it true.

Here are a few basic rules for exponents to remember:

• \(a^m \times a^n = a^{m+n}\)
• \((a^m)^n = a^{mn}\)
• \(a^n / a^m = a^{n-m}\)
• \(a^0 = 1\), as long as \(a eq 0\)

Grasping these rules can greatly ease solving equations involving exponents.

Equations

Equations are mathematical statements that express the equality between two expressions. The original problem, \(12^{x-3} = 1\), is an example of an equation where the goal is to find the value of \(x\) that satisfies it. To solve such equations, you often look for patterns or properties that simplify them, like recognizing \(12^0 = 1\).

When solving equations:

• Identify any mathematical properties that might simplify the expression. Here, recognizing the zero exponent property was key.
• Rewrite the equation using these properties. In this case, \(x-3 = 0\) was derived from the idea that any base to zero power is 1.
• Proceed to solve the simpler equation, like adding 3 to both sides to find \(x\).

This process reveals the value of the variable that balances the equation and provides the solution.

Algebra

Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations. It allows for solving real-world problems and theoretical constructs through manipulation of these symbols. In the exercise given, the expression \(12^{x-3} = 1\) used algebraic concepts to manipulate and solve for \(x\).

Here’s a simple breakdown of how algebra helps:

• Start by rewriting complex expressions using known mathematical properties, such as exponents or logarithms.
• Isolate the variable you’re solving for, in this case \(x\), to one side of the equation using inverse operations, like addition or subtraction.
• Ensure logical steps are followed to maintain the balance of the equation, leading towards the solution.

Algebra provides the tools needed to rearrange and solve equations systematically, revealing solutions that are not immediately apparent.