Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each \(x\)-value is a solution of the equation. \(2^{3 x+1}=128\) (a) \(x=-1\) (b) \(x=2\)

Step-by-Step Solution

Answer

For the given equation \(2^{3x+1}=128\), \(x=-1\) is not a solution, but \(x=2\) is a solution.

Step 1: Understand the equation and the exponential function

Analyze the equation \(2^{3x+1}=128\). Notice that it's an exponential function where the base is 2. The number 128 is also a power of 2. Specifically, \(128 = 2^7\).

Step 2: Substitute \(x=-1\) into the equation

Substitute the given \(x\) value into the equation to see if it works: \(2^{3(-1)+1}=2^{-3+1}=2^{-2}=0.25.\) Because 0.25 is different from 128, \(x=-1\) is not a solution to the equation.

Step 3: Substitute \(x=2\) into the equation

Now substitute the second \(x\) value into the equation: \(2^{3(2)+1}=2^7=128.\) Since 128 equals 128, \(x=2\) is a solution to the equation.

Key Concepts

Solving Exponential Equations

When addressing the topic of solving exponential equations, it's important to recognize that these equations have variables in the exponents, like in the equation \(2^{3x+1}=128\). The first step in solving is to express both sides of the equation with the same base if possible, as indicated in our example where 128 is re-expressed as \(2^7\). This reveals a more straightforward equation of \(2^{3x+1}=2^7\).

Once the bases are aligned, the exponents on each side must be equal for the equation to hold true due to the One-to-One Property of Exponents. This means that \(3x+1=7\) can be solved with basic algebra to find \(x\). Moreover, inspecting the exponents carefully can save time; if they are not matching or cannot be made alike, such as when different bases cannot be expressed using a common base, alternative strategies might be needed, such as using logarithms.

Exponential Functions

The term exponential functions refers to functions of the form \(f(x) = a^{bx+c}\), where \(a\) is a positive constant, and \(b\) and \(c\) are coefficients affecting the rate of growth or decay and the vertical shift, respectively. In the exercise \(2^{3x+1}\), \(2\) is the base, signifying that the function represents exponential growth since the base is greater than 1.

An important characteristic of exponential functions is that they increase (or decrease) by common factors over equal intervals. For instance, every time \(x\) increases by 1, \(2^{3x+1}\) multiplies by a factor of \(2^3\). Recognizing patterns like this can sometimes provide shortcuts to solving exponential equations without resorting to algebraic manipulation.

Substituting Values in Equations

The act of substituting values in equations is a fundamental skill that involves replacing the variable with a specific value to determine if the equation holds true. This step is critical when checking potential solutions to equations, just as shown in the steps of our exercise. For \(x=-1\), the substitution \(2^{3(-1)+1}\) simplifies to \(2^{-2}\) which equals 0.25, indicating that \(x=-1\) is not a solution.

For \(x=2\), substituting into \(2^{3(2)+1}\) gives us \(2^7\), which accurately equals 128. This demonstrates that \(x=2\) is indeed a solution. When substituting, ensure every occurrence of the variable is replaced, and the equation is simplified correctly, using rules of exponents and arithmetic to arrive at a definitive conclusion about whether the substitution yields a true statement.