Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Do not use a calculator. $$2^{3-x}=8$$

Step-by-Step Solution

Answer

x = 0

Step 1: Recognize the Base

The equation given is \(2^{3-x} = 8\). Recognize that 8 can be expressed with the same base as the term on the left. Here, 8 can be rewritten as \(2^3\) since \(2^3 = 8\).

Step 2: Substituting Equivalent Exponent

Rewrite the equation using the base expression for 8 from Step 1: \(2^{3-x} = 2^3\)

Step 3: Equate the Exponents

Since the bases are the same (both are base 2), we can set the exponents equal to each other to solve for \(x\):\( 3 - x = 3\)

Step 4: Solve for x

Subtract 3 from both sides of the equation:\(3 - x - 3 = 3 - 3\)This simplifies to:\(-x = 0\)Multiply by -1 to isolate \(x\):\(x = 0\)

Key Concepts

Base Recognition

Understanding the base in an exponential equation is crucial.
It makes solving these types of problems much simpler. In the equation \[2^{3-x} = 8\]we first need to ensure that both sides of the equation have the same base.
Recognizing that the number 8 can be expressed as a power of 2 is the key step here. By expressing 8 as \[2^3\]we have transformed the equation so both sides share the same base.
This transformation is essential because it allows us to compare directly the exponents of equal bases.
Without base recognition, moving forward to solve the equation might seem difficult or unclear.

Here's the logical pathway:

• First, identify the base in the exponential term.
• Then, find an equivalent base expression for the constant on the other side, if possible.

Exponentiation

Exponentiation is the mathematical operation involving the raising of one number (the base) to the power of another number (the exponent).
In our problem, we look at expressions like \[2^{3-x}\]and \[2^3\]where 2 is the base.
Exponentiation simplifies repetitive multiplication.

• The expression \[2^3\]means multiplying 2 by itself three times: \[2 \times 2 \times 2 = 8\]
• Furthermore, when comparing exponents, if two powers have the same base, we can equate their exponents if the overall expressions are equal.

This is crucial when solving equations that involve exponentiation, as understanding the role each part plays helps you manipulate and solve the problem efficiently.
In our given problem, this understanding allows us to align the exponents once we have consistent bases, making the solution straightforward.

Equation Solving

Once you've recognized and matched the bases, solving the exponential equation becomes a problem of finding the appropriate exponent.
For our problem, we set the equation:\[2^{3-x} = 2^3\]Here, since both sides have the identical base (2), we can directly set their exponents equal to each other:\[3 - x = 3\]The task now is to solve this equation for \(x\).
Here's how we do it step-by-step:

• Subtract 3 from both sides: \[3 - x - 3 = 3 - 3\]This simplifies to:
• \[-x = 0\]Now, multiply both sides by -1 to solve for \(x\):\[x = 0\]

Solving equations by aligning exponents after equalizing bases reduces the complexity of the problem to a simple arithmetic operation.
This structured approach ensures accuracy and builds a foundation for solving more complicated exponential equations in the future.