Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Do not use a calculator. $$3^{5-x}=1$$

Step-by-Step Solution

Answer

The value of \( x \) is 5.

Step 1: Identify the Equation to Solve

The equation to solve is given as \( 3^{5-x} = 1 \). Our aim is to find the value of \( x \) that makes this equation true.

Step 2: Recognize the Power of Zero Law

Remember that any non-zero number raised to the power of zero equals one, i.e., \( a^0 = 1 \) for any \( a eq 0 \). In our problem, this means that we need the exponent on the base 3 to be zero.

Step 3: Set the Exponent to Zero

Since \( 3^{5-x} \) must equal \( 1 \), set the exponent \( 5-x \) equal to 0 based on the previous step: \[ 5 - x = 0 \]

Step 4: Solve for x

To find the value of \( x \), we solve the equation: \[ 5 - x = 0 \] Add \( x \) to both sides to isolate the variable: \[ 5 = x \]

Key Concepts

Power of Zero Law

This is a fundamental rule in mathematics which states that any non-zero base raised to the power of zero equals one. In mathematical terms, if \( a \) is not equal to zero, then \( a^0 = 1 \). This principle stems from the properties of exponents and is essential for solving equations where the result is one. When you see a problem like \( 3^{5-x} = 1 \), the power of zero law tells us immediately that \( 5-x \) must equal zero, because that is the only exponent that results in a base of 3 being equal to one. Understanding this rule simplifies solving the equation drastically, allowing us to set the exponent directly to zero and solve for the variable.

Solving Equations

Solving equations is a key aspect of mathematics that involves finding the unknown values that make the equation true. In the equation \( 3^{5-x} = 1 \), the unknown is \( x \). To solve it, we leverage the power of zero law to simplify our work.

• First, identify the structure of the problem, here an exponential equation.
• Use known mathematical laws to transform the equation into a simpler form.
• In this problem, we apply the power of zero law, setting \( 5-x \) to zero.
• Once the exponent is set to zero, we dismiss the complex part—dealing with the exponential function—and move to simpler algebra.

Ultimately, solving this equation is about transforming a complex-looking problem into a straightforward one where basic arithmetic can be applied.

Basic Algebra

Basic algebra provides the foundation for manipulating equations and solving for unknown variables. After reducing our equation \( 3^{5-x} = 1 \) to \( 5-x = 0 \), we delve into basic algebra to find the value of \( x \). Here are the steps:

• We start with the equation \( 5-x = 0 \).
• Add \( x \) to both sides to isolate the variable.
• You get \( 5 = x \), showing that the solution is simple and intuitive.

Basic algebra requires understanding operations such as addition, subtraction, and the properties of equality, which allow for the systematic solving of equations. By following these orderly steps, we can solve equations step by step, even those stemming from more complex exponential forms.