Solve The Equation: ${ 4^{2x-1} = 14 \cdot 91 }$

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Introduction

Solving equations involving exponents can be a challenging task, especially when dealing with variables in the exponent. In this article, we will focus on solving the equation $4^{2x-1} = 14 \cdot 91$. This equation involves a base of 4 raised to a power of $2x-1$, and we need to find the value of $x$ that satisfies this equation.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand what it means. The equation $4^{2x-1} = 14 \cdot 91$ states that the value of $4$ raised to the power of $2x-1$ is equal to the product of $14$ and $91$. To simplify this equation, we can rewrite $14$ and $91$ as powers of $2$ and $7$, respectively.

Simplifying the Equation

We can rewrite $14$ as $2 \cdot 7$ and $91$ as $7^2$. Substituting these values into the equation, we get:

$$4^{2x-1} = 2 \cdot 7^3$$

Using Properties of Exponents

Now that we have simplified the equation, we can use the properties of exponents to rewrite the equation in a more manageable form. We can rewrite $4$ as $2^2$, and then use the property of exponents that states $(a^m)^n = a^{mn}$.

$$2^{4x-2} = 2 \cdot 7^3$$

Isolating the Variable

Now that we have rewritten the equation, we can isolate the variable $x$ by dividing both sides of the equation by $2^4$. This will give us:

$$2^{4x-6} = 7^3$$

Using Logarithms

To solve for $x$, we can use logarithms to bring the exponent down. We can take the logarithm of both sides of the equation, and then use the property of logarithms that states $\log_a (b^c) = c \log_a b$.

$$\log_2 (2^{4x-6}) = \log_2 (7^3)$$

Simplifying the Logarithmic Equation

Now that we have taken the logarithm of both sides of the equation, we can simplify the equation by using the property of logarithms that states $\log_a (a^c) = c$. This will give us:

$$4x-6 = 3 \log_2 7$$

Solving for x

Now that we have simplified the equation, we can solve for $x$ by adding $6$ to both sides of the equation and then dividing both sides by $4$.

$$4x = 3 \log_2 7 + 6$$

$$x = \frac{3 \log_2 7 + 6}{4}$$

Conclusion

In this article, we have solved the equation $4^{2x-1} = 14 \cdot 91$ using properties of exponents and logarithms. We have rewritten the equation in a more manageable form, isolated the variable $x$, and then used logarithms to bring the exponent down. Finally, we have solved for $x$ by adding $6$ to both sides of the equation and then dividing both sides by $4$. The solution to the equation is $x = \frac{3 \log_2 7 + 6}{4}$.

Additional Tips and Tricks

• When solving equations involving exponents, it's often helpful to rewrite the equation in a more manageable form by using properties of exponents.
• Using logarithms can be a powerful tool for solving equations involving exponents.
• When solving for $x$, be sure to isolate the variable and then use algebraic manipulations to solve for $x$.

Frequently Asked Questions

• Q: What is the value of $x$ that satisfies the equation $4^{2x-1} = 14 \cdot 91$? A: The value of $x$ that satisfies the equation is $x = \frac{3 \log_2 7 + 6}{4}$.
• Q: How do I solve equations involving exponents? A: To solve equations involving exponents, it's often helpful to rewrite the equation in a more manageable form by using properties of exponents, and then use logarithms to bring the exponent down.
• Q: What are some common mistakes to avoid when solving equations involving exponents? A: Some common mistakes to avoid when solving equations involving exponents include failing to isolate the variable, and not using logarithms to bring the exponent down.

References

• [1] "Exponents and Logarithms" by Math Open Reference
• [2] "Solving Equations Involving Exponents" by Khan Academy
• [3] "Logarithms and Exponents" by Wolfram MathWorld
  

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Q&A: Solving Equations Involving Exponents

Q: What is the value of $x$ that satisfies the equation $4^{2x-1} = 14 \cdot 91$?

A: The value of $x$ that satisfies the equation is $x = \frac{3 \log_2 7 + 6}{4}$.

Q: How do I solve equations involving exponents?

A: To solve equations involving exponents, it's often helpful to rewrite the equation in a more manageable form by using properties of exponents, and then use logarithms to bring the exponent down.

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

A: Some common mistakes to avoid when solving equations involving exponents include failing to isolate the variable, and not using logarithms to bring the exponent down.

Q: Can I use a calculator to solve equations involving exponents?

A: Yes, you can use a calculator to solve equations involving exponents. However, it's often helpful to understand the underlying math and use logarithms to bring the exponent down.

Q: How do I rewrite an equation involving exponents in a more manageable form?

A: To rewrite an equation involving exponents in a more manageable form, you can use properties of exponents such as $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. Logarithmic equations can be used to solve exponential equations.

Q: Can I use a graphing calculator to solve equations involving exponents?

A: Yes, you can use a graphing calculator to solve equations involving exponents. However, it's often helpful to understand the underlying math and use logarithms to bring the exponent down.

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

A: To check your answer when solving an equation involving exponents, you can plug your answer back into the original equation and verify that it is true.

Q: What are some real-world applications of solving equations involving exponents?

A: Solving equations involving exponents has many real-world applications, including finance, science, and engineering. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

Q: Can I use a computer program to solve equations involving exponents?

A: Yes, you can use a computer program such as Mathematica or Maple to solve equations involving exponents. However, it's often helpful to understand the underlying math and use logarithms to bring the exponent down.

Q: How do I choose the right logarithmic base when solving an equation involving exponents?

A: When choosing the right logarithmic base when solving an equation involving exponents, you should consider the base of the exponent and choose a logarithmic base that is convenient for the problem.

Q: What are some common logarithmic bases used in solving equations involving exponents?

A: Some common logarithmic bases used in solving equations involving exponents include base 2, base 10, and base e.

Q: Can I use a table of logarithms to solve equations involving exponents?

A: Yes, you can use a table of logarithms to solve equations involving exponents. However, it's often helpful to understand the underlying math and use logarithms to bring the exponent down.

Q: How do I use a calculator to solve a logarithmic equation?

A: To use a calculator to solve a logarithmic equation, you can enter the equation into the calculator and use the logarithmic function to solve for the variable.

Q: What are some common mistakes to avoid when using a calculator to solve a logarithmic equation?

A: Some common mistakes to avoid when using a calculator to solve a logarithmic equation include entering the equation incorrectly, and not using the logarithmic function to solve for the variable.

Additional Resources

• [1] "Exponents and Logarithms" by Math Open Reference
• [2] "Solving Equations Involving Exponents" by Khan Academy
• [3] "Logarithms and Exponents" by Wolfram MathWorld
• [4] "Calculator Tips and Tricks" by Calculator.org
• [5] "Logarithmic Equations" by Mathway.com

Conclusion

Solving equations involving exponents can be a challenging task, but with the right tools and techniques, it can be done. In this article, we have discussed how to solve equations involving exponents using properties of exponents and logarithms. We have also provided answers to frequently asked questions and additional resources for further learning.