Solve The Equation:$\[ 27^{5x - 4} = 81^{3x} \\]

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving the equation $27^{5x - 4} = 81^{3x}$, which involves manipulating exponential expressions and using logarithmic properties to isolate the variable.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be written in the form $a^x = b$, where $a$ and $b$ are constants, and $x$ is the variable. In the given equation, $27^{5x - 4} = 81^{3x}$, we have two exponential expressions with different bases and exponents.

Manipulating Exponential Expressions

To solve the equation, we need to manipulate the exponential expressions to have the same base. We can rewrite $27$ as $3^3$ and $81$ as $3^4$. This gives us:

$$\left(3^3\right)^{5x - 4} = \left(3^4\right)^{3x}$$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify the expressions:

$$3^{3(5x - 4)} = 3^{4(3x)}$$

Equating Exponents

Since the bases are the same, we can equate the exponents:

$$3(5x - 4) = 4(3x)$$

Expanding and simplifying the equation, we get:

$$15x - 12 = 12x$$

Isolating the Variable

To isolate the variable, we need to get all the terms involving $x$ on one side of the equation. Subtracting $12x$ from both sides gives us:

$$3x - 12 = 0$$

Adding $12$ to both sides gives us:

$$3x = 12$$

Solving for x

Finally, we can solve for $x$ by dividing both sides by $3$:

$$x = \frac{12}{3}$$

$$x = 4$$

Conclusion

In this article, we solved the exponential equation $27^{5x - 4} = 81^{3x}$ by manipulating exponential expressions, equating exponents, and isolating the variable. The solution to the equation is $x = 4$. This problem demonstrates the importance of understanding exponential equations and using logarithmic properties to solve them.

Tips and Tricks

• When solving exponential equations, it's essential to have a deep understanding of algebraic manipulations and properties of exponents.
• Use logarithmic properties to simplify exponential expressions and isolate the variable.
• Be careful when equating exponents, as it's essential to have the same base on both sides of the equation.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decay
• Calculating compound interest and investment returns
• Analyzing data and making predictions in fields like medicine, economics, and social sciences

Final Thoughts

Solving exponential equations requires a combination of algebraic manipulations, logarithmic properties, and a deep understanding of mathematical concepts. By following the steps outlined in this article, you can solve complex exponential equations and apply them to real-world problems.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations Solver
• Wolfram Alpha: Exponential Equations Calculator

Frequently Asked Questions

Q: What is an exponential equation? A: An exponential equation is an equation that involves variables in the exponent, written in the form $a^x = b$.

Q: How do I solve an exponential equation? A: To solve an exponential equation, you need to manipulate the exponential expressions to have the same base, equate the exponents, and isolate the variable.

Q: What are some real-world applications of exponential equations? A: Exponential equations have numerous real-world applications, including modeling population growth and decay, calculating compound interest and investment returns, and analyzing data and making predictions in fields like medicine, economics, and social sciences.

Introduction

Exponential equations can be a challenging topic for many students, but with practice and patience, they can become a powerful tool for solving complex problems. In this article, we will answer some of the most frequently asked questions about exponential equations, providing a comprehensive guide to help you understand and master this important mathematical concept.

Q&A: Exponential Equations

Q: What is an exponential equation?

A: An exponential equation is an equation that involves variables in the exponent, written in the form $a^x = b$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to manipulate the exponential expressions to have the same base, equate the exponents, and isolate the variable. This may involve using logarithmic properties, such as the logarithm of a product or the logarithm of a quotient.

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

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

• Not having the same base on both sides of the equation
• Not equating the exponents correctly
• Not isolating the variable correctly
• Not checking the solution for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original equation and verify that it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

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

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

• Modeling population growth and decay
• Calculating compound interest and investment returns
• Analyzing data and making predictions in fields like medicine, economics, and social sciences

Q: How do I use logarithmic properties to solve exponential equations?

A: To use logarithmic properties to solve exponential equations, you need to apply the logarithm to both sides of the equation, using the properties of logarithms to simplify the expression. This may involve using the logarithm of a product or the logarithm of a quotient.

Q: What are some common logarithmic properties used to solve exponential equations?

A: Some common logarithmic properties used to solve exponential equations include:

• The logarithm of a product: $\log(ab) = \log(a) + \log(b)$
• The logarithm of a quotient: $\log(\frac{a}{b}) = \log(a) - \log(b)$
• The logarithm of a power: $\log(a^b) = b\log(a)$

Q: How do I use the change of base formula to solve exponential equations?

A: To use the change of base formula to solve exponential equations, you need to apply the formula to both sides of the equation, using the properties of logarithms to simplify the expression. The change of base formula is given by:

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

Q: What are some common mistakes to avoid when using the change of base formula?

A: Some common mistakes to avoid when using the change of base formula include:

• Not using the correct base for the logarithm
• Not simplifying the expression correctly
• Not checking the solution for extraneous solutions

Conclusion

Exponential equations can be a challenging topic, but with practice and patience, they can become a powerful tool for solving complex problems. By understanding the properties of logarithms and using the change of base formula, you can solve exponential equations with confidence. Remember to check for extraneous solutions and to use the correct base for the logarithm.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations Solver
• Wolfram Alpha: Exponential Equations Calculator

Final Thoughts

Solving exponential equations requires a combination of algebraic manipulations, logarithmic properties, and a deep understanding of mathematical concepts. By following the steps outlined in this article, you can solve complex exponential equations and apply them to real-world problems.

Frequently Asked Questions

Q: What is an exponential equation? A: An exponential equation is an equation that involves variables in the exponent, written in the form $a^x = b$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve an exponential equation? A: To solve an exponential equation, you need to manipulate the exponential expressions to have the same base, equate the exponents, and isolate the variable.

Q: What are some real-world applications of exponential equations? A: Exponential equations have numerous real-world applications, including modeling population growth and decay, calculating compound interest and investment returns, and analyzing data and making predictions in fields like medicine, economics, and social sciences.

Q: How do I use logarithmic properties to solve exponential equations? A: To use logarithmic properties to solve exponential equations, you need to apply the logarithm to both sides of the equation, using the properties of logarithms to simplify the expression.

Q: What are some common logarithmic properties used to solve exponential equations? A: Some common logarithmic properties used to solve exponential equations include the logarithm of a product, the logarithm of a quotient, and the logarithm of a power.

Q: How do I use the change of base formula to solve exponential equations? A: To use the change of base formula to solve exponential equations, you need to apply the formula to both sides of the equation, using the properties of logarithms to simplify the expression.

Q: What are some common mistakes to avoid when using the change of base formula? A: Some common mistakes to avoid when using the change of base formula include not using the correct base for the logarithm, not simplifying the expression correctly, and not checking the solution for extraneous solutions.