Solve The Following Equation:$\[ 3^2 - 4\left(3^x\right) + 3 = 0 \\]

Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. These equations can be challenging to solve, but with the right approach, they can be tackled. In this article, we will focus on solving the equation $3^2 - 4\left(3^x\right) + 3 = 0$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is $3^2 - 4\left(3^x\right) + 3 = 0$. This equation involves an exponential function with base 3 and an unknown exponent x. The equation can be rewritten as $9 - 4\left(3^x\right) + 3 = 0$.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by combining like terms. We can rewrite the equation as $12 - 4\left(3^x\right) = 0$.

Step 2: Isolate the Exponential Term

Next, we need to isolate the exponential term on one side of the equation. We can do this by subtracting 12 from both sides of the equation, which gives us $-4\left(3^x\right) = -12$.

Step 3: Divide by -4

Now, we need to divide both sides of the equation by -4 to isolate the exponential term. This gives us $\left(3^x\right) = 3$.

Step 4: Use Logarithms to Solve for x

Since the equation involves an exponential function, we can use logarithms to solve for x. We can take the logarithm of both sides of the equation, which gives us $x = \log_3(3)$.

Step 5: Simplify the Logarithmic Expression

The logarithmic expression $\log_3(3)$ can be simplified to 1, since the logarithm of a number to its own base is always 1.

Step 6: Check the Solution

Finally, we need to check our solution by plugging it back into the original equation. If the equation holds true, then our solution is correct.

Conclusion

Solving exponential equations can be challenging, but with the right approach, they can be tackled. In this article, we solved the equation $3^2 - 4\left(3^x\right) + 3 = 0$ by simplifying the equation, isolating the exponential term, dividing by -4, using logarithms to solve for x, simplifying the logarithmic expression, and checking the solution. We hope that this article has provided a clear and concise guide to solving exponential equations.

Tips and Tricks

• When solving exponential equations, it's essential to isolate the exponential term on one side of the equation.
• Use logarithms to solve for x when the equation involves an exponential function.
• Simplify the logarithmic expression to get the final solution.
• Check the solution by plugging it back into the original equation.

Common Mistakes to Avoid

• Not isolating the exponential term on one side of the equation.
• Not using logarithms to solve for x when the equation involves an exponential function.
• Not simplifying the logarithmic expression to get the final solution.
• Not checking the solution by plugging it back into the original equation.

Real-World Applications

Exponential equations have many real-world applications, including:

• Modeling population growth and decline
• Calculating compound interest
• Analyzing the spread of diseases
• Predicting the behavior of complex systems

Conclusion

Introduction

In our previous article, we discussed how to solve exponential equations using a step-by-step approach. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will provide a Q&A guide to help you better understand how to solve exponential equations.

Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. Exponential functions are functions that involve a base raised to a power, such as 2^x or 3^x.

Q: How do I know if an equation is exponential?

A: An equation is exponential if it involves an exponential function, such as 2^x or 3^x. You can also check if the equation involves a base raised to a power.

Q: What is the first step in solving an exponential equation?

A: The first step in solving an exponential equation is to simplify the equation by combining like terms.

Q: How do I isolate the exponential term?

A: To isolate the exponential term, you need to get rid of any constants or other terms that are not exponential. You can do this by subtracting or adding the same value to both sides of the equation.

Q: What is the next step after isolating the exponential term?

A: After isolating the exponential term, you need to use logarithms to solve for x. This involves taking the logarithm of both sides of the equation.

Q: What type of logarithm should I use?

A: You should use a logarithm that is the same base as the exponential function. For example, if the exponential function is 2^x, you should use a base 2 logarithm.

Q: How do I simplify the logarithmic expression?

A: To simplify the logarithmic expression, you need to use the properties of logarithms. For example, if you have log(a) + log(b), you can simplify it to log(ab).

Q: What is the final step in solving an exponential equation?

A: The final step in solving an exponential equation is to check your solution by plugging it back into the original equation.

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 using logarithms to solve for x
• Not simplifying the logarithmic expression
• Not checking the solution

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

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

• Modeling population growth and decline
• Calculating compound interest
• Analyzing the spread of diseases
• Predicting the behavior of complex systems

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises. You can also try solving real-world problems that involve exponential equations.

Conclusion

In conclusion, solving exponential equations requires a clear and concise approach. By following the steps outlined in this article, you can solve exponential equations with ease. Remember to isolate the exponential term, use logarithms to solve for x, simplify the logarithmic expression, and check the solution. With practice and patience, you can become proficient in solving exponential equations and apply them to real-world problems.