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Introduction

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Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will focus on solving the equation $2^{4x} = 8^{3x - 10}$, which involves manipulating exponential expressions and using logarithmic properties to isolate the variable.

Understanding Exponential Equations

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Exponential equations involve variables raised to a power, and they can be written in the form $a^x = b^y$, where $a$ and $b$ are constants, and $x$ and $y$ are variables. To solve these equations, we need to use the properties of exponents, such as the product rule, the quotient rule, and the power rule.

The Product Rule

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The product rule states that when we multiply two exponential expressions with the same base, we can add their exponents. Mathematically, this can be represented as:

$$a^x \cdot a^y = a^{x+y}$$

The Quotient Rule

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The quotient rule states that when we divide two exponential expressions with the same base, we can subtract their exponents. Mathematically, this can be represented as:

$$\frac{a^x}{a^y} = a^{x-y}$$

The Power Rule

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The power rule states that when we raise an exponential expression to a power, we can multiply the exponent by the power. Mathematically, this can be represented as:

$$(a^x)^y = a^{xy}$$

Solving the Equation

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Now that we have a good understanding of the properties of exponents, let's focus on solving the equation $2^{4x} = 8^{3x - 10}$.

Step 1: Rewrite the Equation

The first step in solving the equation is to rewrite it in a more manageable form. We can rewrite $8$ as $2^3$, so the equation becomes:

$$2^{4x} = (2^3)^{3x - 10}$$

Step 2: Apply the Power Rule

Using the power rule, we can rewrite the equation as:

$$2^{4x} = 2^{3(3x - 10)}$$

Step 3: Simplify the Exponents

Now that we have the same base on both sides of the equation, we can equate the exponents:

$$4x = 3(3x - 10)$$

Step 4: Distribute the 3

Distributing the 3 on the right-hand side of the equation, we get:

$$4x = 9x - 30$$

Step 5: Subtract 4x from Both Sides

Subtracting $4x$ from both sides of the equation, we get:

$$0 = 5x - 30$$

Step 6: Add 30 to Both Sides

Adding 30 to both sides of the equation, we get:

$$30 = 5x$$

Step 7: Divide Both Sides by 5

Dividing both sides of the equation by 5, we get:

$$6 = x$$

Conclusion

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In this article, we have solved the exponential equation $2^{4x} = 8^{3x - 10}$ using the properties of exponents and logarithmic properties. We have shown that the solution to the equation is $x = 6$. This demonstrates the importance of understanding the properties of exponents and logarithms in solving complex mathematical equations.

Final Answer

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The final answer is $\boxed{6}$.

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Introduction

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In our previous article, we solved the exponential equation $2^{4x} = 8^{3x - 10}$ using the properties of exponents and logarithmic properties. 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&A: Solving Exponential Equations

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable raised to a power, such as $a^x = b^y$, where $a$ and $b$ are constants, and $x$ and $y$ are variables.

Q: What are the properties of exponents?

A: The properties of exponents include the product rule, the quotient rule, and the power rule. The product rule states that when we multiply two exponential expressions with the same base, we can add their exponents. The quotient rule states that when we divide two exponential expressions with the same base, we can subtract their exponents. The power rule states that when we raise an exponential expression to a power, we can multiply the exponent by the power.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to use the properties of exponents and logarithmic properties to isolate the variable. Here are the steps:

1. Rewrite the equation in a more manageable form.
2. Apply the power rule to simplify the exponents.
3. Equate the exponents on both sides of the equation.
4. Solve for the variable.

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

A: An exponential equation involves a variable raised to a power, while a logarithmic equation involves a variable that is the exponent of a number. For example, $2^x = 8$ is an exponential equation, while $x = \log_2 8$ is a logarithmic equation.

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

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

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

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

• Not rewriting the equation in a more manageable form.
• Not applying the power rule correctly.
• Not equating the exponents on both sides of the equation.
• Not solving for the variable correctly.

Example Problems

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Problem 1

Solve the equation $3^x = 9^{2x - 1}$.

Solution

1. Rewrite the equation in a more manageable form: $3^x = (3^2)^{2x - 1}$.
2. Apply the power rule: $3^x = 3^{4x - 2}$.
3. Equate the exponents: $x = 4x - 2$.
4. Solve for the variable: $3x = 2$, $x = \frac{2}{3}$.

Problem 2

Solve the equation $2^{3x} = 8^{x - 2}$.

Solution

1. Rewrite the equation in a more manageable form: $2^{3x} = (2^3)^{x - 2}$.
2. Apply the power rule: $2^{3x} = 2^{3x - 6}$.
3. Equate the exponents: $3x = 3x - 6$.
4. Solve for the variable: $6 = 0$, which is a contradiction.

Conclusion

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In this article, we have provided a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations. We have also provided example problems to help you practice solving exponential equations. Remember to always rewrite the equation in a more manageable form, apply the power rule correctly, and equate the exponents on both sides of the equation.

Final Tips

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• Always use the properties of exponents and logarithmic properties to simplify the equation and isolate the variable.
• Be careful when applying the power rule and equating the exponents.
• Practice solving exponential equations to become more confident and proficient.

Final Answer

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The final answer is $\boxed{\frac{2}{3}}$.