Calculating the Value of x

Oct 13, 2025 by talynt.org 54 views

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 a specific exponential equation and provide a step-by-step guide on how to do it.

The Exponential Equation

The exponential equation we will be solving is:

${ 12 = 10^{x+3} - 9 }$

This equation involves an exponential function with a base of 10 and an exponent of x+3. Our goal is to isolate the variable x and find its value.

Step 1: Add 9 to Both Sides

To start solving the equation, we need to get rid of the negative term on the right-hand side. We can do this by adding 9 to both sides of the equation:

${ 12 + 9 = 10^{x+3} - 9 + 9 }$

This simplifies to:

${ 21 = 10^{x+3} }$

Step 2: Take the Logarithm of Both Sides

Next, we need to get rid of the exponential function. We can do this by taking the logarithm of both sides of the equation. We will use the natural logarithm (ln) for this purpose:

${ \ln(21) = \ln(10^{x+3}) }$

Using the property of logarithms that states $\ln(a^b) = b\ln(a)$ , we can simplify the right-hand side of the equation:

${ \ln(21) = (x+3)\ln(10) }$

Step 3: Divide Both Sides by ln(10)

To isolate the variable x, we need to get rid of the coefficient of x. We can do this by dividing both sides of the equation by ln(10):

${ \frac{\ln(21)}{\ln(10)} = x+3 }$

Step 4: Subtract 3 from Both Sides

Finally, we need to isolate the variable x. We can do this by subtracting 3 from both sides of the equation:

${ x = \frac{\ln(21)}{\ln(10)} - 3 }$

Calculating the Value of x

Now that we have the equation for x, we can calculate its value using a calculator:

${ x = \frac{\ln(21)}{\ln(10)} - 3 }$

${ x \approx \frac{3.044}{2.302} - 3 }$

${ x \approx 1.323 - 3 }$

${ x \approx -1.677 }$

Conclusion

In this article, we solved an exponential equation using a step-by-step approach. We started by adding 9 to both sides of the equation, then took the logarithm of both sides, divided both sides by ln(10), and finally subtracted 3 from both sides. The final answer was x ≈ -1.677.

Answer Key

The correct answer is:

A. -1.677

Discussion

This problem is a great example of how to solve exponential equations. The key is to use the properties of logarithms to get rid of the exponential function and then isolate the variable x. If you have any questions or need further clarification, please don't hesitate to ask.

References

[1] "Exponential Functions" by Math Is Fun
[2] "Logarithmic Functions" by Math Is Fun
[3] "Solving Equations" by Khan Academy

Additional Resources

[1] Exponential and Logarithmic Functions by MIT OpenCourseWare
[2] Algebra by Khan Academy
[3] Mathway: Algebra Solver
Exponential Equation Q&A ==========================

Introduction

In our previous article, we solved an exponential equation using a step-by-step approach. In this article, we will answer some frequently asked questions about exponential equations and provide additional resources for further learning.

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 10^x.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable x. You can do this by using the properties of logarithms to get rid of the exponential function and then solving for x.

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

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

Not using the correct properties of logarithms
Not isolating the variable x
Not checking the domain of the function
Not using a calculator to check the answer

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

A: To check your answer when solving an exponential equation, you need to plug your solution back into the original equation and make sure it is true. You can also use a calculator to check your answer.

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

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

Modeling population growth
Modeling chemical reactions
Modeling financial growth
Modeling electrical circuits

Q: How do I graph an exponential function?

A: To graph an exponential function, you need to use a graphing calculator or a computer program. You can also use a table of values to create a graph.

Q: What are some common exponential functions?

A: Some common exponential functions include:

2^x
10^x
e^x
a^x

Q: How do I find the inverse of an exponential function?

A: To find the inverse of an exponential function, you need to swap the x and y variables and then solve for y.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

Use the properties of logarithms to get rid of the exponential function
Isolate the variable x
Check the domain of the function
Use a calculator to check the answer

Conclusion

In this article, we answered some frequently asked questions about exponential equations and provided additional resources for further learning. We hope this article has been helpful in understanding exponential equations and how to solve them.

Additional Resources

[1] Exponential and Logarithmic Functions by MIT OpenCourseWare
[2] Algebra by Khan Academy
[3] Mathway: Algebra Solver
[4] Exponential Functions by Math Is Fun
[5] Logarithmic Functions by Math Is Fun

Discussion

This article is a great resource for anyone who wants to learn more about exponential equations and how to solve them. If you have any questions or need further clarification, please don't hesitate to ask.

References

[1] "Exponential Functions" by Math Is Fun
[2] "Logarithmic Functions" by Math Is Fun
[3] "Solving Equations" by Khan Academy

Answer Key

Q1: An exponential equation is a type of mathematical equation that involves an exponential function.
Q2: To solve an exponential equation, you need to isolate the variable x using the properties of logarithms.
Q3: Some common mistakes to avoid when solving exponential equations include not using the correct properties of logarithms, not isolating the variable x, not checking the domain of the function, and not using a calculator to check the answer.
Q4: To check your answer when solving an exponential equation, you need to plug your solution back into the original equation and make sure it is true.
Q5: Exponential equations have many real-world applications, including modeling population growth, modeling chemical reactions, modeling financial growth, and modeling electrical circuits.
Q6: To graph an exponential function, you need to use a graphing calculator or a computer program.
Q7: Some common exponential functions include 2^x, 10^x, e^x, and a^x.
Q8: To find the inverse of an exponential function, you need to swap the x and y variables and then solve for y.
Q9: Some tips for solving exponential equations include using the properties of logarithms to get rid of the exponential function, isolating the variable x, checking the domain of the function, and using a calculator to check the answer.