Find The Solution Of The Exponential Equation, Correct To Three Decimal Places.${ 5^{-\frac{x}{1300}} = 3 }$ { X = \}

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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 with ease. In this article, we will focus on solving the exponential equation $5^{-\frac{x}{1300}} = 3$ and provide a step-by-step guide on how to find the solution.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential function, which is a function that raises a number to a power. The general form of an exponential equation is $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result. Exponential equations can be solved using various methods, including logarithmic methods and algebraic methods.

The Given Equation

The given equation is $5^{-\frac{x}{1300}} = 3$. This equation involves an exponential function with a base of 5 and an exponent of $-\frac{x}{1300}$. The result of the equation is 3.

Step 1: Isolate the Exponent

To solve the equation, we need to isolate the exponent. We can do this by taking the logarithm of both sides of the equation. Since the base of the exponential function is 5, we can use the logarithm base 5 to isolate the exponent.

Step 2: Take the Logarithm of Both Sides

Taking the logarithm base 5 of both sides of the equation, we get:

$$\log_5(5^{-\frac{x}{1300}}) = \log_5(3)$$

Using the property of logarithms that states $\log_a(a^x) = x$, we can simplify the left-hand side of the equation:

$$-\frac{x}{1300} = \log_5(3)$$

Step 3: Solve for x

Now that we have isolated the exponent, we can solve for $x$. To do this, we can multiply both sides of the equation by $-1300$:

$$x = -1300 \log_5(3)$$

Step 4: Evaluate the Logarithm

To evaluate the logarithm, we can use a calculator or a logarithmic table. The logarithm base 5 of 3 is approximately 0.238.

Step 5: Calculate the Value of x

Now that we have the value of the logarithm, we can calculate the value of $x$:

$$x = -1300 \times 0.238$$

$$x = -311.4$$

Conclusion

In this article, we have solved the exponential equation $5^{-\frac{x}{1300}} = 3$ using a step-by-step approach. We have isolated the exponent, taken the logarithm of both sides, and solved for $x$. The value of $x$ is approximately $-311.4$, correct to three decimal places.

Final Answer

The final answer is $\boxed{-311.4}$.

Additional Tips and Resources

• To solve exponential equations, it is essential to understand the properties of logarithms and how to use them to isolate the exponent.
• When solving exponential equations, it is crucial to check the validity of the solution by plugging it back into the original equation.
• For more information on exponential equations and logarithms, refer to the following resources:
  • Khan Academy: Exponential and Logarithmic Equations
  • Mathway: Exponential and Logarithmic Equations
  • Wolfram Alpha: Exponential and Logarithmic Equations

Common Mistakes to Avoid

• When solving exponential equations, it is easy to make mistakes by not isolating the exponent correctly or by not checking the validity of the solution.
• To avoid these mistakes, it is essential to follow the steps outlined in this article and to double-check the solution by plugging it back into the original equation.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating interest rates and investments
• Analyzing data and trends in various fields, such as economics, biology, and physics

Conclusion

In conclusion, solving exponential equations requires a step-by-step approach and a good understanding of logarithmic properties. By following the steps outlined in this article, you can solve exponential equations with ease and apply the concepts to real-world problems.

Introduction

Solving exponential equations can be a challenging task, but with the right approach and understanding of the concepts, it can be done with ease. In this article, we will answer some of the most frequently asked questions about solving exponential equations.

Q1: What is an exponential equation?

A1: An exponential equation is a type of mathematical equation that involves an exponential function. The general form of an exponential equation is $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q2: How do I solve an exponential equation?

A2: To solve an exponential equation, you need to isolate the exponent. You can do this by taking the logarithm of both sides of the equation. Since the base of the exponential function is $a$, you can use the logarithm base $a$ to isolate the exponent.

Q3: What is the logarithm of an exponential equation?

A3: The logarithm of an exponential equation is a way to isolate the exponent. It is a mathematical operation that finds the power to which a base must be raised to produce a given value.

Q4: How do I use logarithms to solve exponential equations?

A4: To use logarithms to solve exponential equations, you need to take the logarithm of both sides of the equation. This will allow you to isolate the exponent and solve for $x$.

Q5: What is the difference between a logarithmic equation and an exponential equation?

A5: A logarithmic equation is an equation that involves a logarithmic function, while an exponential equation is an equation that involves an exponential function. Logarithmic equations are used to solve exponential equations.

Q6: How do I check the validity of a solution to an exponential equation?

A6: To check the validity of a solution to an exponential equation, you need to plug the solution back into the original equation and verify that it is true.

Q7: What are some common mistakes to avoid when solving exponential equations?

A7: Some common mistakes to avoid when solving exponential equations include not isolating the exponent correctly, not checking the validity of the solution, and not using the correct logarithm base.

Q8: How do I apply exponential equations to real-world problems?

A8: Exponential equations can be applied to real-world problems in various fields, including economics, biology, and physics. For example, exponential equations can be used to model population growth and decline, calculate interest rates and investments, and analyze data and trends.

Q9: What are some resources for learning more about exponential equations?

A9: Some resources for learning more about exponential equations include Khan Academy, Mathway, and Wolfram Alpha. These resources provide step-by-step instructions and examples for solving exponential equations.

Q10: Can I use a calculator to solve exponential equations?

A10: Yes, you can use a calculator to solve exponential equations. However, it is essential to understand the concepts and methods involved in solving exponential equations, as using a calculator alone may not provide a complete understanding of the problem.

Conclusion

In conclusion, solving exponential equations requires a good understanding of logarithmic properties and a step-by-step approach. By following the steps outlined in this article and using the resources provided, you can solve exponential equations with ease and apply the concepts to real-world problems.

Additional Tips and Resources

• To learn more about exponential equations, refer to the following resources:
  • Khan Academy: Exponential and Logarithmic Equations
  • Mathway: Exponential and Logarithmic Equations
  • Wolfram Alpha: Exponential and Logarithmic Equations
• For more information on logarithmic properties and how to use them to solve exponential equations, refer to the following resources:
  • Khan Academy: Logarithmic Properties
  • Mathway: Logarithmic Properties
  • Wolfram Alpha: Logarithmic Properties

Common Mistakes to Avoid

• When solving exponential equations, it is easy to make mistakes by not isolating the exponent correctly or by not checking the validity of the solution.
• To avoid these mistakes, it is essential to follow the steps outlined in this article and to double-check the solution by plugging it back into the original equation.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decline
• Calculating interest rates and investments
• Analyzing data and trends in various fields, such as economics, biology, and physics

Conclusion

In conclusion, solving exponential equations requires a good understanding of logarithmic properties and a step-by-step approach. By following the steps outlined in this article and using the resources provided, you can solve exponential equations with ease and apply the concepts to real-world problems.