Choose The Logarithmic Form Of The Exponential Equation $3^7 = (4x + 3)$.A. Log ⁡ 7 3 = ( 4 X + 3 \log_7 3 = (4x + 3 Lo G 7 ​ 3 = ( 4 X + 3 ] B. Log ⁡ 7 ( 4 X + 3 ) = 3 \log_7(4x + 3) = 3 Lo G 7 ​ ( 4 X + 3 ) = 3 C. Log ⁡ 3 7 = ( 4 X + 3 \log_3 7 = (4x + 3 Lo G 3 ​ 7 = ( 4 X + 3 ] D. Log ⁡ 3 ( 4 X + 3 ) = 7 \log_3(4x + 3) = 7 Lo G 3 ​ ( 4 X + 3 ) = 7

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic functions. In this article, we will explore the logarithmic form of the exponential equation $3^7 = (4x + 3)$. We will examine the different options and determine the correct logarithmic form.

Understanding Exponential Equations

Exponential equations involve a base raised to a power, and the result is equal to a specific value. In the given equation, $3^7 = (4x + 3)$, the base is 3, and the power is 7. The result is equal to $4x + 3$.

Logarithmic Form

To solve exponential equations, we can use logarithmic form. The logarithmic form of an exponential equation is written as $\log_b a = c$, where $b$ is the base, $a$ is the result, and $c$ is the power.

Option A: $\log_7 3 = (4x + 3)$

Option A is incorrect because the base of the logarithm is 7, but the result is $4x + 3$, not 3. The correct logarithmic form should have the base as the result, not the other way around.

Option B: $\log_7(4x + 3) = 3$

Option B is also incorrect because the power of the logarithm is 3, but the result of the exponential equation is $4x + 3$, not 7. The correct logarithmic form should have the base as the result, not the other way around.

Option C: $\log_3 7 = (4x + 3)$

Option C is incorrect because the base of the logarithm is 3, but the result is $4x + 3$, not 7. The correct logarithmic form should have the base as the result, not the other way around.

Option D: $\log_3(4x + 3) = 7$

Option D is the correct logarithmic form of the exponential equation $3^7 = (4x + 3)$. The base of the logarithm is 3, the result is $4x + 3$, and the power is 7.

Conclusion

In conclusion, the correct logarithmic form of the exponential equation $3^7 = (4x + 3)$ is $\log_3(4x + 3) = 7$. This is because the base of the logarithm is 3, the result is $4x + 3$, and the power is 7.

Step-by-Step Solution

To solve the exponential equation $3^7 = (4x + 3)$, we can use the following steps:

1. Write the equation in logarithmic form: $\log_3(4x + 3) = 7$
2. Use the definition of logarithms to rewrite the equation: $3^7 = 4x + 3$
3. Solve for $x$: $4x = 3^7 - 3$
4. Simplify the: $4x = 2187 - 3$
5. Solve for $x$: $4x = 2184$
6. Divide both sides by 4: $x = \frac{2184}{4}$
7. Simplify the equation: $x = 546$

Final Answer

The final answer is $\boxed{546}$.

Additional Resources

For more information on logarithmic form and exponential equations, please refer to the following resources:

• Khan Academy: Logarithmic Form
• Mathway: Exponential Equations
• Wolfram Alpha: Logarithmic Form

Conclusion

Q: What is the logarithmic form of an exponential equation?

A: The logarithmic form of an exponential equation is written as $\log_b a = c$, where $b$ is the base, $a$ is the result, and $c$ is the power.

Q: How do I determine the correct logarithmic form of an exponential equation?

A: To determine the correct logarithmic form of an exponential equation, you need to identify the base, the result, and the power. The base should be the result, and the power should be the exponent.

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

A: The logarithmic form of an equation is written as $\log_b a = c$, while the exponential form is written as $b^c = a$. The logarithmic form is used to solve for the exponent, while the exponential form is used to solve for the result.

Q: How do I solve an exponential equation using logarithmic form?

A: To solve an exponential equation using logarithmic form, you need to follow these steps:

1. Write the equation in logarithmic form: $\log_b a = c$
2. Use the definition of logarithms to rewrite the equation: $b^c = a$
3. Solve for the variable: $x = \frac{a}{b^c}$

Q: What is the base of the logarithm in the logarithmic form of an exponential equation?

A: The base of the logarithm in the logarithmic form of an exponential equation is the result of the exponential equation.

Q: What is the power of the logarithm in the logarithmic form of an exponential equation?

A: The power of the logarithm in the logarithmic form of an exponential equation is the exponent of the exponential equation.

Q: Can I use logarithmic form to solve exponential equations with different bases?

A: Yes, you can use logarithmic form to solve exponential equations with different bases. However, you need to make sure that the base of the logarithm is the same as the base of the exponential equation.

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

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

• Using the wrong base or power
• Not following the correct order of operations
• Not simplifying the equation properly

Q: How do I check my answer when solving an exponential equation using logarithmic form?

A: To check your answer when solving an exponential equation using logarithmic form, you need to plug the solution back into the original equation and make sure that it is true.

Conclusion

In conclusion, the logarithmic form of an exponential equation is a powerful tool for solving exponential equations. By following the correct steps and avoiding common mistakes, you can use logarithmic form to solve exponential equations with different bases. Remember to check your answer carefully to ensure that it is correct.

Additional Resources

For more information on logarithmic form and exponential equations, please refer to the following resources:

• Khan Academy: Logarithmic Form
• Mathway: Exponential Equations
• Wolfram Alpha: Logarithmic Form

Final Answer

The final answer is $\boxed{546}$.