Solve For $x$ To The Nearest Tenth.$900 = 720(1.06)^{x+2} - 3500$\$x = \square$[/tex\]

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulation and logarithmic properties. In this article, we will focus on solving a specific exponential equation, which involves isolating the variable x to the nearest tenth. The given equation is:

900=720(1.06)x+2−3500900 = 720(1.06)^{x+2} - 3500

Our goal is to solve for x and provide a clear, step-by-step solution.

Step 1: Isolate the Exponential Term

To begin solving the equation, we need to isolate the exponential term on one side of the equation. We can do this by adding 3500 to both sides of the equation:

900+3500=720(1.06)x+2900 + 3500 = 720(1.06)^{x+2}

This simplifies to:

4400=720(1.06)x+24400 = 720(1.06)^{x+2}

Step 2: Divide Both Sides by 720

Next, we need to divide both sides of the equation by 720 to isolate the exponential term:

4400720=(1.06)x+2\frac{4400}{720} = (1.06)^{x+2}

This simplifies to:

6.1111=(1.06)x+26.1111 = (1.06)^{x+2}

Step 3: Take the Logarithm of Both Sides

To solve for x, we need to take the logarithm of both sides of the equation. We can use any base for the logarithm, but in this case, we will use the natural logarithm (ln):

ln⁡(6.1111)=ln⁡((1.06)x+2)\ln(6.1111) = \ln((1.06)^{x+2})

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

ln⁡(6.1111)=(x+2)ln⁡(1.06)\ln(6.1111) = (x+2)\ln(1.06)

Step 4: Simplify the Equation

Now, we can simplify the equation by evaluating the natural logarithms:

1.8053=(x+2)(0.0582)1.8053 = (x+2)(0.0582)

Step 5: Solve for x

To solve for x, we need to isolate x on one side of the equation. We can do this by dividing both sides of the equation by 0.0582:

1.80530.0582=x+2\frac{1.8053}{0.0582} = x+2

This simplifies to:

31.0=x+231.0 = x+2

Step 6: Subtract 2 from Both Sides

Finally, we need to subtract 2 from both sides of the equation to solve for x:

31.0−2=x31.0 - 2 = x

This simplifies to:

x=29.0x = 29.0

Conclusion

In this article, we solved a specific exponential equation by isolating the variable x to the nearest tenth. We used a step-by-step approach, including isolating the exponential term, dividing both sides by 720, taking the logarithm of both sides, simplifying the equation, and solving for x. The final solution is x = 29.0.

Tips and Tricks

  • When solving exponential equations, it's essential to isolate the exponential term on one side of the equation.
  • Use logarithms to simplify the equation and solve for the variable.
  • Be careful when evaluating logarithms and simplifying the equation.
  • Always check your solution by plugging it back into the original equation.

Common Mistakes

  • Failing to isolate the exponential term on one side of the equation.
  • Not using logarithms to simplify the equation.
  • Not checking 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 compound interest and investment returns.
  • Analyzing the spread of diseases and epidemics.
  • Predicting the behavior of complex systems, such as financial markets and climate models.

Introduction

In our previous article, we solved a specific exponential equation by isolating the variable x to the nearest tenth. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a mathematical expression that involves a base raised to a power. Exponential equations can be written in the form:

ax=ba^x = b

or

ax+b=ca^x + b = c

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term on one side of the equation. You can do this by using algebraic manipulation, such as adding or subtracting terms, and then using logarithms to simplify the equation.

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

A: A logarithmic equation is an equation that involves a logarithmic expression, which is the inverse of an exponential expression. Logarithmic equations can be written in the form:

log⁡a(x)=y\log_a(x) = y

or

log⁡a(x)+y=z\log_a(x) + y = z

Exponential equations, on the other hand, involve an exponential expression, which is a mathematical expression that involves a base raised to a power.

Q: How do I use logarithms to solve an exponential equation?

A: To use logarithms to solve an exponential equation, you need to take the logarithm of both sides of the equation. This will allow you to simplify the equation and isolate the variable.

Q: What is the most common type of exponential equation?

A: The most common type of exponential equation is the equation of the form:

ax=ba^x = b

This type of equation is often used to model population growth and decline, as well as to calculate compound interest and investment returns.

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug it back into the original equation and verify that it is true. This will ensure that your solution is correct and accurate.

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

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

  • Failing to isolate the exponential term on one side of the equation.
  • Not using logarithms to simplify the equation.
  • Not checking the solution by plugging it back into the original equation.

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

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

  • Modeling population growth and decline.
  • Calculating compound interest and investment returns.
  • Analyzing the spread of diseases and epidemics.
  • Predicting the behavior of complex systems, such as financial markets and climate models.

By understanding how to solve exponential equations, you can apply these concepts to a wide range of real-world problems and make informed decisions.

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 optimization problems.
  • Using exponential equations to analyze data and make predictions.

By mastering these advanced topics, you can take your understanding of exponential equations to the next level and apply them to even more complex and challenging problems.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations. By mastering these concepts, you can apply exponential equations to a wide range of real-world problems and make informed decisions.