Problem 71 - Algebra 2

Question

Solve each equation. $$ 3^{2 x}=6561 $$

Step-by-Step Solution

Answer

The solution to the equation \(3^{2x} = 6561\) is \(x = 4\).

Step 1: Rewrite the given number into a base of 3

Recognize that \(6561 = 3^8\). So the equation \(3^{2x} = 6561\) can be rewritten as \(3^{2x} = 3^8\).

Step 2: Comparison of Exponents

This is the stage where the most crucial realization is made. Since both sides of the equation are now in terms of the same base, the only way the equation can be true is if the exponents are equal. Therefore, set \(2x = 8\). At that moment, the original exponential equation has been transformed into a simple linear equation.

Step 3: Solve for x

Now you have a simple linear equation, solve it to find x. Divide both sides of the equation \(2x = 8\) by 2 to obtain \(x = 4\).

Key Concepts

Exponential Functions

Exponential functions are essential mathematical expressions where a constant base is raised to a variable exponent. They are represented in the general form, \(f(x) = a^x\), where \(a\) is the base and \(x\) is the exponent. These functions have distinct properties:

• The base \(a\) is a positive number.
• The function grows rapidly due to the variable exponent.
• The graph of an exponential function is a smooth curve increasing or decreasing swiftly.

Exponential functions are used in numerous real-world applications, such as population growth, radioactivity, and investment finance. Understanding them lays a foundation for comprehending complex mathematical models and problem-solving techniques.

Exponents

Exponents are a fundamental concept in algebra, representing a number multiplying itself a certain number of times. In the expression \(a^b\), \(a\) is known as the base and \(b\) as the exponent. Here's how exponents work:

• If \(b = 2\), it means \(a\) is multiplied by itself once: \(a \times a = a^2\).
• Exponents can be positive, negative, or zero.
• Common rules include the product of powers rule \(a^m \times a^n = a^{m+n}\) and the power of a power rule \((a^m)^n = a^{m \cdot n}\).

By manipulating exponents, we can simplify complex expressions and solve equations efficiently. Understanding exponent rules is vital for success in algebra and higher-level mathematics.

Linear Equations

Linear equations are algebraic equations where the highest exponent of the variable is one. They form straight line graphs when plotted, signifying a constant rate of change. The general form is \(ax + b = 0\), where \(a\) and \(b\) are constants, and \(x\) is the variable. Solving linear equations involves:

• Isolating the variable on one side of the equation.
• Performing arithmetic to solve for the variable.

In the context of transforming an exponential equation to a linear one, as seen in the exercise, it's essential to understand that once bases are equal, the problem simplifies drastically. Solving linear equations is a critical skill necessary for algebra students to master.

Bases in Algebra

In algebra, the base is the number or expression that is repeated in multiplication, as governed by the exponent. Understanding this concept helps in tackling exponential equations efficiently. The base affects how quickly the exponential function grows:

• When the base \(a > 1\), the function describes exponential growth.
• Conversely, \(0 < a < 1\) results in exponential decay.
• Changing the base changes the form and behavior of the function, thereby affecting the graph's shape.

Recognizing the base allows you to rewrite complex expressions into simpler terms, as demonstrated in the problem where non-standard expressions are converted to familiar base forms. Mastery of base manipulation is a useful tool in the algebra toolbox.