Problem 58 - Beginning and Intermediate Algebra

Question

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$2^{x^{2}-9 x}=\frac{1}{256}$$

Step-by-Step Solution

Answer

The logarithmic equation \(2^{x^2 - 9x} = \frac{1}{256}\) can be solved algebraically by taking the base 2 logarithm of both sides, applying logarithm properties, and solving for x: \((x^2 - 9x) = -8\) \(x^2 - 9x + 8 = 0\) \((x - 1)(x - 8) = 0\) This yields two possible solutions: \(x = 1\) and \(x = 8\). Checking these solutions using a graphing calculator confirms their validity, as the graphs of \(y = 2^{x^2 - 9x}\) and \(y = \frac{1}{256}\) intersect at points (1, 1/256) and (8, 1/256).

Step 1: Take the logarithm of both sides of the equation

We need to take the logarithm of both sides of the equation. We'll use the base 2 logarithm, as it will cancel out the base 2 exponent on the left side of the equation. \(log_2(2^{x^2 - 9x}) = log_2\left(\frac{1}{256}\right)\)

Step 2: Apply the logarithm properties to simplify the equation

Use the properties of logarithms to simplify the equation: Property 1: \(log_a(a^b) = b\) Property 2: \(log_a\left(\frac{1}{a^b}\right)=-b\) Applying Property 1 to the left side of the equation: \((x^2 - 9x) = log_2\left(\frac{1}{256}\right)\) Now, applying Property 2 to the right side of the equation: \((x^2 - 9x) = -8\)

Step 3: Solve for x algebraically

To solve for x, first rearrange the equation: \(x^2 - 9x + 8 = 0\) Now, we'll try to factor the quadratic: \((x - 1)(x - 8) = 0\) So, we have two possible solutions: \(x - 1 = 0\) => \(x = 1\) \(x - 8 = 0\) => \(x = 8\)

Step 4: Check the solution(s) using a graphing calculator

Using a graphing calculator, we can check our solutions by graphing the two functions: \(y = 2^{x^2 - 9x}\) (Original graph) \(y = \frac{1}{256}\) (Horizontal line as a reference) We can observe that the graphs intersect at points (1, 1/256) and (8, 1/256), confirming that our calculated solutions x = 1 and x = 8 are correct.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to the power of a variable or expression. An example of an exponential function is \(2^{x^2 - 9x}\). Here, the function has a base of 2 and an exponent of \(x^2 - 9x\). These functions are widely used in various fields such as finance, science, and engineering for modeling growth and decay processes.

• Graphing: Exponential functions typically produce curved graphs that show exponential growth or decay. As the exponent in the function increases, the value of the function either grows rapidly (in the case of growth) or decays towards zero (in the case of decay).
• Base: The base of an exponential function is crucial. A base greater than 1 generally indicates growth, while a base between 0 and 1 signifies decay. In this exercise, the base is 2, which suggests growth if not for the negative impact of the exponent.
• Transformation: Exponential functions can be transformed through various operations, such as translations and reflections, that affect the function's graph and behavior.

Understanding the nature of exponential functions and their behaviors unlocks a powerful tool for exploring and interpreting mathematical relationships that change rapidly.

Logarithm Properties

Logarithm properties can simplify and solve equations involving exponential functions, like the one in this exercise. Logarithms are the inverses of exponential functions and help translate multiplicative relationships into additive ones, which are often easier to solve.

For this problem, we applied two specific logarithm properties:

• Power Rule: \(\log_a(a^b) = b\) - This property simplifies any expression where the logarithm base equals the exponential base. By applying it, the exponent becomes the answer to the logarithm.
• Inverse Rule: \(\log_a\left(\frac{1}{a^b}\right) = -b\) - This is useful when dealing with reciprocals. It states that the logarithm of the reciprocal of a power is the negative of the exponent.

By using these properties, we converted the complex exponential equation into a much simpler polynomial equation in terms of \(x^2 - 9x = -8\). Understanding and applying logarithm properties is key to solving such algebraic problems quickly and accurately.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In step 3 of solving the logarithmic equation, we transformed the expression \(x^2 - 9x + 8 = 0\) into a quadratic equation.

• Factoring: In this case, we factored the quadratic equation into two binomials, \((x - 1)(x - 8) = 0\). Factoring helps to discover the roots or solutions of the equation, where each factor represents a point where the function crosses the \(x\)-axis.
• Solutions: The solutions to the quadratic equation are \(x = 1\) and \(x = 8\). These are the points where the original exponential equation \(2^{x^2 - 9x} = \frac{1}{256}\) holds true.
• Verifying Solutions: It's a good practice to verify solutions using tools like graphing calculators. By confirming that the intersections are accurate, we ensure that the solutions meet both the algebraic and graphical requirements.

Solving quadratic equations efficiently is essential in many fields, as they frequently appear in various scientific and engineering contexts.