To Solve $49^{3x} = 343^{2x+1}$, Write Each Side Of The Equation In Terms Of A Common Base.

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Introduction

When solving equations involving exponents, it's often helpful to express both sides of the equation in terms of a common base. This allows us to easily compare the exponents and solve for the variable. In this article, we'll show you how to rewrite the equation $49^{3x} = 343^{2x+1}$ in terms of a common base and solve for the variable x.

Understanding the Equation

The given equation is $49^{3x} = 343^{2x+1}$. To begin solving this equation, we need to understand the properties of exponents and how to rewrite expressions in terms of a common base.

Rewriting the Equation in Terms of a Common Base

We can rewrite the equation by expressing both sides in terms of a common base. Since both 49 and 343 are powers of 7, we can rewrite the equation as follows:

493x=(72)3x=76x49^{3x} = (7^2)^{3x} = 7^{6x}

3432x+1=(73)2x+1=76x+3343^{2x+1} = (7^3)^{2x+1} = 7^{6x+3}

Now that we have rewritten the equation in terms of a common base, we can easily compare the exponents and solve for the variable x.

Solving for x

Since both sides of the equation are now expressed in terms of the common base 7, we can set the exponents equal to each other and solve for x:

6x=6x+36x = 6x+3

Subtracting 6x from both sides gives us:

0=30 = 3

This is a contradiction, which means that the original equation has no solution.

Conclusion

In this article, we showed you how to rewrite the equation $49^{3x} = 343^{2x+1}$ in terms of a common base and solve for the variable x. By expressing both sides of the equation in terms of a common base, we were able to easily compare the exponents and determine that the original equation has no solution.

Common Bases and Exponents

When working with exponents, it's often helpful to express expressions in terms of a common base. This allows us to easily compare the exponents and solve for the variable. In this section, we'll discuss some common bases and how to rewrite expressions in terms of these bases.

The Number 7 as a Common Base

The number 7 is a common base for many expressions involving exponents. For example, we can rewrite the expression $49^{3x}$ in terms of the base 7 as follows:

493x=(72)3x=76x49^{3x} = (7^2)^{3x} = 7^{6x}

Similarly, we can rewrite the expression $343^{2x+1}$ in terms of the base 7 as follows:

3432x+1=(73)2x+1=76x+3343^{2x+1} = (7^3)^{2x+1} = 7^{6x+3}

The Number 10 as a Common Base

The number 10 is also a common base for many expressions involving exponents. For example, we can rewrite the expression $100^{3x}$ in terms of the base 10 as follows:

1003x=(102)3x=106x100^{3x} = (10^2)^{3x} = 10^{6x}

Similarly, we can rewrite the expression $1000^{2x+1}$ in terms of the base 10 as follows:

10002x+1=(103)2x+1=106x+31000^{2x+1} = (10^3)^{2x+1} = 10^{6x+3}

Tips and Tricks

When working with exponents, it's often helpful to express expressions in terms of a common base. Here are some tips and tricks to keep in mind:

  • Use a common base: When working with exponents, it's often helpful to express expressions in terms of a common base. This allows us to easily compare the exponents and solve for the variable.
  • Rewrite expressions: Rewrite expressions in terms of a common base by using the properties of exponents. For example, we can rewrite the expression $49^3x}$ in terms of the base 7 as follows $49^{3x = (72){3x} = 7^{6x}$
  • Compare exponents: Once we have rewritten the expressions in terms of a common base, we can easily compare the exponents and solve for the variable.

Final Thoughts

In this article, we showed you how to rewrite the equation $49^{3x} = 343^{2x+1}$ in terms of a common base and solve for the variable x. By expressing both sides of the equation in terms of a common base, we were able to easily compare the exponents and determine that the original equation has no solution. We also discussed some common bases and how to rewrite expressions in terms of these bases. Finally, we provided some tips and tricks for working with exponents and rewriting expressions in terms of a common base.

Introduction

In our previous article, we showed you how to rewrite the equation $49^{3x} = 343^{2x+1}$ in terms of a common base and solve for the variable x. However, we received many questions from readers who were struggling to understand the concept of solving equations with exponents. In this article, we'll answer some of the most frequently asked questions about solving equations with exponents.

Q: What is the difference between a base and an exponent?

A: A base is the number that is being raised to a power, while an exponent is the power to which the base is being raised. For example, in the expression $2^3$, the base is 2 and the exponent is 3.

Q: How do I rewrite an expression in terms of a common base?

A: To rewrite an expression in terms of a common base, you need to identify the base and the exponent. Then, you can use the properties of exponents to rewrite the expression in terms of the common base. For example, to rewrite the expression $49^{3x}$ in terms of the base 7, you can use the property $a^{mn} = (am)n$ to get $49^{3x} = (72){3x} = 7^{6x}$.

Q: How do I compare exponents when the bases are different?

A: When the bases are different, you can compare the exponents by using the property $a^m = b^n \Rightarrow m = n \log_b a$. This property allows you to compare the exponents by taking the logarithm of both sides of the equation.

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. For example, the equation $\log_2 x = 3$ is a logarithmic equation, while the equation $2^x = 8$ is an exponential equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the property $\log_b a = c \Rightarrow b^c = a$. This property allows you to rewrite the logarithmic equation as an exponential equation, which can then be solved using the properties of exponents.

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

A: Some common mistakes to avoid when solving equations with exponents include:

  • Not rewriting the expression in terms of a common base: Failing to rewrite the expression in terms of a common base can make it difficult to compare the exponents and solve the equation.
  • Not using the properties of exponents: Failing to use the properties of exponents can make it difficult to rewrite the expression in terms of a common base and compare the exponents.
  • Not checking for extraneous solutions: Failing to check for extraneous solutions can result in incorrect solutions to the equation.

Q: What are some tips for solving equations with exponents?

A: Some tips for solving equations with exponents include:

  • Use a common base: Using a common base can make it easier to compare the exponents and solve the equation.
  • Rewrite the expression in terms of a common base: Rewriting the expression in terms of a common base can make it easier to compare the exponents and solve the equation.
  • Use the properties of exponents: Using the properties of exponents can make it easier to rewrite the expression in terms of a common base and compare the exponents.
  • Check for extraneous solutions: Checking for extraneous solutions can help ensure that the solutions to the equation are correct.

Conclusion

In this article, we answered some of the most frequently asked questions about solving equations with exponents. We discussed the difference between a base and an exponent, how to rewrite an expression in terms of a common base, and how to compare exponents when the bases are different. We also discussed the difference between a logarithmic and exponential equation, how to solve a logarithmic equation, and some common mistakes to avoid when solving equations with exponents. Finally, we provided some tips for solving equations with exponents.