What Values Of B Satisfy 3 2b 3 2 36

What Values of b Satisfy 3^(2b) = 3^(2) * 3^(6)? Solving Exponential Equations

This article delves into the solution of the exponential equation 3^(2b) = 3^(2) * 3^(6), exploring the underlying mathematical principles and demonstrating a step-by-step solution process. We'll examine various approaches to solving this type of problem and discuss the significance of understanding exponential properties. Beyond the immediate solution, we'll consider broader implications and applications of similar equations in various fields.

Understanding Exponential Properties

Before we dive into solving the equation, let's refresh our understanding of key exponential properties. These properties are crucial for manipulating and simplifying exponential expressions:

• Product of Powers: When multiplying two exponential expressions with the same base, we add the exponents: a^m * aⁿ = a^(m+n). This is fundamental to simplifying the right-hand side of our equation.

• Power of a Power: When raising an exponential expression to a power, we multiply the exponents: (a^m)ⁿ = a^(m*n). While not directly used in this specific problem, understanding this rule is crucial for solving more complex exponential equations.

• Equality of Exponents: If a^m = aⁿ, and 'a' is a positive number not equal to 1, then m = n. This property allows us to equate the exponents and solve for the unknown variable.

Solving the Equation: 3^(2b) = 3^(2) * 3^(6)

Now, let's tackle the equation: 3^(2b) = 3^(2) * 3^(6). Our goal is to find the value(s) of 'b' that satisfy this equation.

Step 1: Simplify the Right-Hand Side

Using the product of powers property, we can simplify the right-hand side of the equation:

3^(2) * 3^(6) = 3^(2+6) = 3^(8)

This simplifies our equation to:

3^(2b) = 3^(8)

Step 2: Equate the Exponents

Since the bases are the same (both are 3), and the base is a positive number not equal to 1, we can equate the exponents:

2b = 8

Step 3: Solve for b

This is a simple algebraic equation. To solve for 'b', we divide both sides of the equation by 2:

b = 8/2 = 4

Therefore, the value of 'b' that satisfies the equation 3^(2b) = 3^(2) * 3^(6) is b = 4.

Verification

To verify our solution, we can substitute b = 4 back into the original equation:

3^(2*4) = 3^(8)

3^(8) = 3^(8)

The equation holds true, confirming that our solution, b = 4, is correct.

Alternative Approaches

While the method above is the most straightforward, let's explore alternative approaches to solving similar exponential equations:

1. Using Logarithms:

For more complex exponential equations, logarithms can be a powerful tool. We can take the logarithm of both sides of the equation:

log(3^(2b)) = log(3^(2) * 3^(6))

Using the logarithm properties, we can simplify:

2b * log(3) = 2 * log(3) + 6 * log(3)

2b * log(3) = 8 * log(3)

Dividing both sides by 2 * log(3) (assuming log(3) ≠ 0), we get:

b = 4

This method confirms our earlier result. However, for this specific equation, the previous method is simpler and more efficient.

2. Graphical Approach:

We can also solve this equation graphically. By plotting the functions y = 3^(2b) and y = 3^(8) on the same graph, the x-coordinate of their intersection point will represent the solution for 'b'. This visual method can be particularly helpful for understanding the behavior of exponential functions and visualizing the solution.

Applications of Exponential Equations

Exponential equations, like the one we solved, have wide-ranging applications across various fields, including:

• Finance: Compound interest calculations rely heavily on exponential functions. Understanding exponential growth allows us to model and predict investment growth over time.

• Biology: Population growth, bacterial growth, and radioactive decay are often modeled using exponential functions.

• Physics: Radioactive decay and the behavior of certain physical phenomena are described by exponential equations.

• Computer Science: The complexity of certain algorithms can be expressed using exponential functions, helping us analyze their efficiency and scalability.

• Chemistry: Chemical reactions, particularly those involving radioactive substances, follow exponential decay patterns.

Expanding on Exponential Concepts

Understanding exponential equations extends beyond simply solving for a single variable. It involves grasping the fundamental concepts of exponential growth and decay, which are crucial for understanding many real-world phenomena. Exponential functions are characterized by their rapid growth or decline, and their properties are essential for modelling and predicting various trends and processes.

The ability to manipulate and simplify exponential expressions, as demonstrated in solving the equation above, is a valuable skill that extends far beyond mathematical exercises. It's a fundamental concept applied across numerous disciplines, and a strong understanding of this topic is a stepping stone to mastering more advanced mathematical concepts.

Furthermore, mastering the skill of solving exponential equations helps to build a solid foundation for tackling more complex mathematical problems involving exponential and logarithmic functions, including calculus problems that involve exponential functions. The skill of utilizing logarithmic properties alongside the properties of exponents are crucial in finding solutions for complex equations.

In conclusion, the solution to the equation 3^(2b) = 3^(2) * 3^(6) is b = 4. This solution, however, serves as more than just an answer; it's a demonstration of fundamental exponential properties and a stepping stone to understanding the broader applications of exponential equations in various fields. By mastering these principles, you equip yourself with valuable tools for solving a wide range of mathematical problems and for understanding the world around us more deeply.