Solve For { Y $} . . . { 3 \cdot 10^{2y} = 3,600 \}

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving exponential equations of the form $a \cdot b^{xy} = c$, where $a$, $b$, $x$, and $y$ are constants. We will use the given equation $3 \cdot 10^{2y} = 3,600$ as a case study to illustrate the step-by-step process of solving exponential equations.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, which can make them challenging to solve. However, with a solid understanding of the underlying principles, we can break down these equations into manageable parts. The key concept to grasp is that exponential functions have a base and an exponent. In the given equation, the base is $10$, and the exponent is $2y$.

Step 1: Isolate the Exponential Term

The first step in solving the equation is to isolate the exponential term. In this case, we can start by dividing both sides of the equation by $3$ to get:

$$10^{2y} = \frac{3,600}{3}$$

This simplifies to:

$$10^{2y} = 1,200$$

Step 2: Express the Right-Hand Side as a Power of 10

The next step is to express the right-hand side of the equation as a power of $10$. We can do this by finding the logarithm of $1,200$ to the base $10$. Using a calculator or a logarithm table, we find that:

$$\log_{10} 1,200 \approx 3.079$$

This means that:

$$10^{3.079} \approx 1,200$$

Step 3: Equate the Exponents

Now that we have expressed the right-hand side as a power of $10$, we can equate the exponents. Since the bases are the same, we can set the exponents equal to each other:

$$2y = 3.079$$

Step 4: Solve for y

The final step is to solve for $y$. We can do this by dividing both sides of the equation by $2$:

$$y = \frac{3.079}{2}$$

This simplifies to:

$$y \approx 1.5395$$

Conclusion

Solving exponential equations requires a deep understanding of the underlying principles and a step-by-step approach. By isolating the exponential term, expressing the right-hand side as a power of $10$, equating the exponents, and solving for $y$, we can arrive at the solution. In this case, we found that $y \approx 1.5395$. This demonstrates the power of exponential equations and the importance of understanding these concepts in mathematics.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Finance: Exponential equations are used to model population growth, compound interest, and inflation.
• Biology: Exponential equations are used to model population growth, disease spread, and chemical reactions.
• Physics: Exponential equations are used to model radioactive decay, electrical circuits, and wave propagation.

Tips and Tricks

Here are some tips and tricks to help you solve exponential equations:

• Use logarithms: Logarithms can help you simplify exponential equations and make them easier to solve.
• Isolate the exponential term: Isolating the exponential term can help you focus on the exponent and make it easier to solve.
• Express the right-hand side as a power of 10: Expressing the right-hand side as a power of 10 can help you equate the exponents and solve for $y$.

Practice Problems

Here are some practice problems to help you reinforce your understanding of exponential equations:

• Problem 1: Solve the equation $2 \cdot 5^{3x} = 1,000$.
• Problem 2: Solve the equation $3 \cdot 2^{4y} = 48$.
• Problem 3: Solve the equation $4 \cdot 3^{2x} = 324$.

Conclusion

$$$$

Frequently Asked Questions

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. It is a mathematical statement that contains a base and an exponent, and the variable is raised to a power.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term, express the right-hand side as a power of 10, equate the exponents, and solve for the variable.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation involves a variable in the exponent, while a linear equation involves a variable in the coefficient. Exponential equations are more complex and require a different approach to solve.

Q: Can I use logarithms to solve exponential equations?

A: Yes, logarithms can be used to solve exponential equations. By taking the logarithm of both sides of the equation, you can simplify the equation and make it easier to solve.

Q: How do I express the right-hand side of an exponential equation as a power of 10?

A: To express the right-hand side of an exponential equation as a power of 10, you need to find the logarithm of the right-hand side to the base 10. This will give you the exponent, which you can then use to equate the exponents.

Q: What is the significance of the base in an exponential equation?

A: The base in an exponential equation is the number that is raised to the power of the variable. The base can be any positive number, but it is usually a prime number or a power of a prime number.

Q: Can I use a calculator to solve exponential equations?

A: Yes, a calculator can be used to solve exponential equations. However, it is always a good idea to check your work and make sure that the solution is correct.

Q: How do I check my work when solving an exponential equation?

A: To check your work, you need to substitute the solution back into the original equation and make sure that it is true. You can also use a calculator to check your work.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not isolating the exponential term
• Not expressing the right-hand side as a power of 10
• Not equating the exponents
• Not checking your work

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have numerous real-world applications, including:

• Finance: Exponential equations are used to model population growth, compound interest, and inflation.
• Biology: Exponential equations are used to model population growth, disease spread, and chemical reactions.
• Physics: Exponential equations are used to model radioactive decay, electrical circuits, and wave propagation.

Conclusion

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. By following the steps outlined in this article, you can become proficient in solving exponential equations and apply these concepts to real-world problems. Remember to practice regularly and check your work to ensure that you are solving the equations correctly.