Solving Exponential Equations 2^(3x-5) = 4

In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, exponential equations hold a significant place. These equations involve variables in the exponents, requiring specific techniques to find solutions. This comprehensive guide delves into the process of solving a particular exponential equation, providing a step-by-step approach and illuminating the underlying principles.

Understanding Exponential Equations

Before we embark on solving the equation, let's first grasp the essence of exponential equations. An exponential equation is an equation in which the variable appears in the exponent. These equations often involve exponents, requiring a deep understanding of exponent rules and properties. The key to solving exponential equations lies in manipulating the equation to isolate the variable. This often involves expressing both sides of the equation with the same base, a technique we will explore in detail.

The Equation at Hand

We are presented with the equation: $2^{3x-5} = 4$. Our mission is to find the value(s) of x that satisfy this equation. This equation exemplifies a typical exponential equation where the unknown, x, resides in the exponent. To solve for x, we'll employ the strategy of expressing both sides of the equation with the same base. This technique simplifies the equation and allows us to equate the exponents, leading to a straightforward algebraic solution.

Step-by-Step Solution

Now, let's embark on the journey of solving the equation, breaking it down into manageable steps:

Step 1: Expressing Both Sides with the Same Base

The cornerstone of solving exponential equations is to express both sides with the same base. In our equation, we have $2^{3x-5}$ on the left side and 4 on the right side. We recognize that 4 can be expressed as a power of 2, specifically $2^2$. Therefore, we rewrite the equation as:

$2^{3x-5} = 2^2$

This transformation is crucial as it allows us to directly compare the exponents. By expressing both sides with the same base, we've paved the way for the next step in our solution.

Step 2: Equating the Exponents

With both sides of the equation expressed with the same base, we can now equate the exponents. This principle stems from the fundamental property of exponential functions: if $a^m = a^n$, then m = n. Applying this to our equation, we have:

$3x - 5 = 2$

We have successfully transformed the exponential equation into a linear equation. This linear equation is much simpler to solve, requiring only basic algebraic manipulation.

Step 3: Solving the Linear Equation

Now, we focus on solving the linear equation $3x - 5 = 2$. To isolate x, we first add 5 to both sides of the equation:

$3x - 5 + 5 = 2 + 5$

This simplifies to:

$3x = 7$

Next, we divide both sides by 3:

$3x / 3 = 7 / 3$

This yields the solution:

$x = 7/3$

We have successfully solved for x, finding that x = 7/3 is the solution to the original exponential equation.

The Solution

Therefore, the solution to the equation $2^{3x-5} = 4$ is:

A. The solution is $x = 7/3$.

Verification

To ensure the correctness of our solution, we can substitute x = 7/3 back into the original equation:

$2^{3(7/3)-5} = 2^{7-5} = 2^2 = 4$

Since the left side equals the right side, our solution is verified.

Conclusion

In this guide, we have meticulously solved the exponential equation $2^{3x-5} = 4$. By expressing both sides with the same base, equating the exponents, and solving the resulting linear equation, we arrived at the solution x = 7/3. This step-by-step approach provides a clear framework for tackling similar exponential equations. Understanding the underlying principles and practicing these techniques will empower you to confidently solve a wide range of exponential equations. Mastering exponential equations is a crucial step in advancing your mathematical skills.

Remember, the key to solving exponential equations is to manipulate the equation strategically, leveraging the properties of exponents to isolate the variable. Practice and persistence are your allies in this mathematical endeavor. As you encounter more exponential equations, you'll develop an intuition for the most effective solution methods. Embrace the challenge and unlock the power of exponential equations!

Additional Tips for Solving Exponential Equations

Here are some additional tips to enhance your problem-solving skills when dealing with exponential equations:

• Simplify Expressions: Before attempting to solve, simplify any expressions on either side of the equation. This might involve combining like terms or applying exponent rules.
• Look for Common Bases: If possible, try to express all terms with the same base. This often simplifies the equation significantly.
• Use Logarithms: If you cannot express both sides with the same base, logarithms can be your friend. Take the logarithm of both sides to bring the variable out of the exponent.
• Check for Extraneous Solutions: When using logarithms, remember to check for extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation.
• Practice Regularly: The more you practice, the more comfortable you will become with solving exponential equations. Work through a variety of problems to solidify your understanding.

The Importance of Exponential Equations

Exponential equations are not just abstract mathematical concepts; they have real-world applications in various fields, including:

• Finance: Compound interest and investment growth are modeled using exponential functions.
• Biology: Population growth and radioactive decay are described by exponential equations.
• Physics: Many physical phenomena, such as the cooling of an object, follow exponential laws.
• Computer Science: Algorithms and data structures often rely on exponential relationships.

By mastering exponential equations, you gain a valuable tool for understanding and modeling the world around you. The ability to solve exponential equations is a testament to your mathematical proficiency and opens doors to further exploration in various scientific and technical domains. Continue your journey in mathematics and discover the power of exponential functions!