Consider $8^{x-4} = 8^{10}$.Because The Bases Are Equal, The Exponents Must Also Be Equal. Solve For X X X .

by ADMIN 111 views

===========================================================

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving exponential equations with the same base, using the given equation $8^{x-4} = 8^{10}$ as an example. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential expression, which is a mathematical expression that represents the result of raising a number to a power. The general form of an exponential equation is $a^x = b$, where aa is the base, xx is the exponent, and bb is the result. In the given equation $8^{x-4} = 8^{10}$, the base is 8, and the exponents are x−4x-4 and 10.

The Rule of Exponents

When solving exponential equations, we use the rule of exponents, which states that if the bases are equal, the exponents must also be equal. This means that if we have two exponential expressions with the same base, we can set the exponents equal to each other and solve for the variable.

Applying the Rule of Exponents

In the given equation $8^{x-4} = 8^{10}$, we can apply the rule of exponents by setting the exponents equal to each other:

x−4=10x-4 = 10

Solving for xx

Now that we have a linear equation, we can solve for xx by adding 4 to both sides of the equation:

x=10+4x = 10 + 4

x=14x = 14

Therefore, the value of xx that satisfies the equation $8^{x-4} = 8^{10}$ is x=14x = 14.

Conclusion

Solving exponential equations with the same base requires a deep understanding of the underlying principles and the rule of exponents. By applying the rule of exponents and solving for the variable, we can find the value of the exponent that satisfies the equation. In this article, we used the given equation $8^{x-4} = 8^{10}$ as an example and solved for xx using the rule of exponents.

Examples and Applications

Exponential equations have numerous applications in various fields, including science, engineering, and economics. Here are a few examples:

  • Population growth: Exponential equations can be used to model population growth, where the population grows at a rate proportional to the current population.
  • Compound interest: Exponential equations can be used to calculate compound interest, where the interest is compounded at a rate proportional to the current balance.
  • Radioactive decay: Exponential equations can be used to model radioactive decay, where the amount of radioactive material decreases at a rate proportional to the current amount.

Tips and Tricks

Here are a few tips and tricks to help you solve exponential equations:

  • Check the bases: Before applying the rule of exponents, make sure that the bases are equal.
  • Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
  • Use algebraic manipulations: Use algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, to isolate the variable.

Practice Problems

Here are a few practice problems to help you practice solving exponential equations:

  • 2x+3=272^{x+3} = 2^{7}

  • 3x−2=393^{x-2} = 3^{9}

  • 4x+1=4114^{x+1} = 4^{11}

Conclusion

Solving exponential equations with the same base requires a deep understanding of the underlying principles and the rule of exponents. By applying the rule of exponents and solving for the variable, we can find the value of the exponent that satisfies the equation. With practice and patience, you can become proficient in solving exponential equations and apply them to real-world problems.

Glossary

Here are a few key terms related to exponential equations:

  • Exponent: A number that represents the power to which a base is raised.
  • Base: A number that is raised to a power.
  • Rule of exponents: A mathematical rule that states that if the bases are equal, the exponents must also be equal.

References

Here are a few references that you can use to learn more about exponential equations:

  • Algebra: A comprehensive textbook on algebra that covers exponential equations and other topics.
  • Mathematics: A textbook on mathematics that covers exponential equations and other topics.
  • Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha that provide interactive lessons and practice problems on 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 represents the result of raising a number to a power. The general form of an exponential equation is $a^x = b$, where aa is the base, xx is the exponent, and bb is the result.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to apply the rule of exponents, which states that if the bases are equal, the exponents must also be equal. This means that if you have two exponential expressions with the same base, you can set the exponents equal to each other and solve for the variable.

Q: What is the rule of exponents?

A: The rule of exponents is a mathematical rule that states that if the bases are equal, the exponents must also be equal. This means that if you have two exponential expressions with the same base, you can set the exponents equal to each other and solve for the variable.

Q: How do I apply the rule of exponents?

A: To apply the rule of exponents, you need to set the exponents equal to each other and solve for the variable. For example, if you have the equation $2^{x+3} = 2^{7}$, you can set the exponents equal to each other and solve for xx:

x+3=7x+3 = 7

x=7−3x = 7 - 3

x=4x = 4

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

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

  • Not checking the bases: Before applying the rule of exponents, make sure that the bases are equal.
  • Not simplifying the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
  • Not using algebraic manipulations: Use algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, to isolate the variable.

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

A: To check if an equation is an exponential equation, look for the following characteristics:

  • Exponential expression: The equation should contain an exponential expression, which is a mathematical expression that represents the result of raising a number to a power.
  • Same base: The bases of the exponential expressions should be equal.
  • Exponents: The exponents of the exponential expressions should be different.

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

A: Yes, you can use a calculator to solve exponential equations. However, make sure to check the calculator's settings and ensure that it is set to the correct mode (e.g., scientific mode) before using it to solve the equation.

Q: How do I graph an exponential equation?

A: To graph an exponential equation, you can use a graphing calculator or a computer program such as Desmos. To graph an exponential equation, follow these steps:

  1. Enter the equation: Enter the exponential equation into the graphing calculator or computer program.
  2. Set the window: Set the window to the correct range and scale.
  3. Graph the equation: Graph the equation using the graphing calculator or computer program.

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

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

  • Population growth: Exponential equations can be used to model population growth, where the population grows at a rate proportional to the current population.
  • Compound interest: Exponential equations can be used to calculate compound interest, where the interest is compounded at a rate proportional to the current balance.
  • Radioactive decay: Exponential equations can be used to model radioactive decay, where the amount of radioactive material decreases at a rate proportional to the current amount.

Q: How do I practice solving exponential equations?

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

  • Practice problems: Practice solving exponential equations using practice problems, such as those found in textbooks or online resources.
  • Online resources: Use online resources, such as Khan Academy, Mathway, and Wolfram Alpha, to practice solving exponential equations.
  • Graphing calculators: Use graphing calculators to visualize and solve exponential equations.

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

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

  • Not setting the window correctly: Make sure to set the window to the correct range and scale.
  • Not graphing the equation correctly: Make sure to graph the equation using the correct mode (e.g., scientific mode).
  • Not checking the graph: Check the graph to ensure that it is accurate and complete.