Solve The Equation. Round To The Nearest Ten-thousandth. 5 4 X − 2 = 120 5^{4x-2} = 120 5 4 X − 2 = 120 X = □ X = \square X = □

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $5^{4x-2} = 120$ and finding the value of $x$. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Exponential Equations

Exponential equations involve a variable in the exponent, and the base is a constant. In this case, the base is 5, and the exponent is $4x-2$. The equation $5^{4x-2} = 120$ can be rewritten as $5^{4x-2} - 120 = 0$. This is a classic example of an exponential equation, and we will use various techniques to solve it.

Step 1: Isolate the Exponential Term

The first step in solving the equation is to isolate the exponential term. We can do this by subtracting 120 from both sides of the equation:

$$5^{4x-2} - 120 = 0$$

$$5^{4x-2} = 120$$

Step 2: Take the Logarithm of Both Sides

To solve for $x$, we need to get rid of the exponent. We can do this by taking the logarithm of both sides of the equation. We will use the natural logarithm (ln) for this purpose:

$$\ln(5^{4x-2}) = \ln(120)$$

Using the property of logarithms that states $\ln(a^b) = b\ln(a)$, we can rewrite the equation as:

$$(4x-2)\ln(5) = \ln(120)$$

Step 3: Simplify the Equation

Now that we have isolated the term with the variable, we can simplify the equation by dividing both sides by $\ln(5)$:

$$4x-2 = \frac{\ln(120)}{\ln(5)}$$

Step 4: Solve for $x$

The final step is to solve for $x$. We can do this by adding 2 to both sides of the equation and then dividing both sides by 4:

$$4x = \frac{\ln(120)}{\ln(5)} + 2$$

$$x = \frac{\frac{\ln(120)}{\ln(5)} + 2}{4}$$

Using a Calculator to Find the Value of $x$

To find the value of $x$, we can use a calculator to evaluate the expression:

$$x = \frac{\frac{\ln(120)}{\ln(5)} + 2}{4}$$

Using a calculator, we get:

$$x \approx 0.5$$

Conclusion

Solving exponential equations can be challenging, but with the right approach, they can be tackled with ease. In this article, we solved the equation $5^{4x-2} = 120$ and found the value of $x$. We broke down the solution into manageable steps, making it easy to follow and understand. By using logarithms and simplifying the equation, we were able to find the value of $x$.

Tips and Tricks

• When solving exponential equations, it's essential to isolate the exponential term first.
• Use logarithms to get rid of the exponent.
• Simplify the equation by dividing both sides by the logarithm of the base.
• Use a calculator to evaluate the expression and find the value of the variable.

Common Mistakes to Avoid

• Not isolating the exponential term first.
• Not using logarithms to get rid of the exponent.
• Not simplifying the equation by dividing both sides by the logarithm of the base.
• Not using a calculator to evaluate the expression and find the value of the variable.

Real-World Applications

Exponential equations have many real-world applications, including:

• Modeling population growth
• Calculating compound interest
• Analyzing chemical reactions
• Solving problems in physics and engineering

By understanding how to solve exponential equations, you can apply this knowledge to a wide range of real-world problems.

Final Thoughts

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Introduction

In our previous article, we explored the world of exponential equations and solved the equation $5^{4x-2} = 120$. We broke down the solution into manageable steps, making it easy to follow and understand. In this article, we will continue to delve into the world of exponential equations and answer some of the most frequently asked questions.

Q&A

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent, and the base is a constant. For example, $5^{4x-2} = 120$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term first. Then, you can use logarithms to get rid of the exponent. Finally, simplify the equation by dividing both sides by the logarithm of the base.

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

A: A logarithmic equation is an equation that involves a variable in the logarithm, and the base is a constant. For example, $\log_5(120) = 4x-2$ is a logarithmic equation. An exponential equation, on the other hand, involves a variable in the exponent, and the base is a constant.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to solve an exponential equation. However, it's essential to understand the underlying math and not just rely on the calculator.

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 first
• Not using logarithms to get rid of the exponent
• Not simplifying the equation by dividing both sides by the logarithm of the base
• Not using a calculator to evaluate the expression and find the value of the variable

Q: How do I apply exponential equations to real-world problems?

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

• Modeling population growth
• Calculating compound interest
• Analyzing chemical reactions
• Solving problems in physics and engineering

Q: Can I use exponential equations to solve problems in finance?

A: Yes, you can use exponential equations to solve problems in finance. For example, you can use exponential equations to calculate compound interest or to model the growth of an investment.

Q: What are some advanced topics in exponential equations?

A: Some advanced topics in exponential equations include:

• Solving systems of exponential equations
• Using exponential equations to model complex systems
• Applying exponential equations to differential equations

Q: How do I practice solving exponential equations?

A: To practice solving exponential equations, you can try the following:

• Start with simple exponential equations and gradually move on to more complex ones
• Use online resources or textbooks to practice solving exponential equations
• Try to apply exponential equations to real-world problems

Conclusion

Solving exponential equations can be challenging, but with the right approach, they can be tackled with ease. By understanding how to solve exponential equations, you can apply this knowledge to a wide range of real-world problems. We hope this article has provided you with a clear understanding of how to solve exponential equations and has inspired you to explore this fascinating topic further.

Tips and Tricks

• When solving exponential equations, it's essential to isolate the exponential term first.
• Use logarithms to get rid of the exponent.
• Simplify the equation by dividing both sides by the logarithm of the base.
• Use a calculator to evaluate the expression and find the value of the variable.

Common Mistakes to Avoid

• Not isolating the exponential term first.
• Not using logarithms to get rid of the exponent.
• Not simplifying the equation by dividing both sides by the logarithm of the base.
• Not using a calculator to evaluate the expression and find the value of the variable.

Real-World Applications

Exponential equations have many real-world applications, including:

• Modeling population growth
• Calculating compound interest
• Analyzing chemical reactions
• Solving problems in physics and engineering

By understanding how to solve exponential equations, you can apply this knowledge to a wide range of real-world problems.

Final Thoughts

Solving exponential equations can be challenging, but with the right approach, they can be tackled with ease. By understanding how to solve exponential equations, you can apply this knowledge to a wide range of real-world problems. We hope this article has provided you with a clear understanding of how to solve exponential equations and has inspired you to explore this fascinating topic further.