What Is The True Solution To $2 \ln E^{\ln 5 X} = 2 \ln 15$?A. $x = 0$B. $x = 3$C. $x = 9$D. $x = 15$

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Introduction

Mathematics is a vast and complex subject that deals with numbers, quantities, and shapes. It involves various branches, including algebra, geometry, calculus, and more. One of the fundamental concepts in mathematics is logarithms, which are used to solve equations and express complex relationships between numbers. In this article, we will explore the true solution to the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$.

Understanding the Equation

The given equation is $2 \ln e^{\ln 5 x} = 2 \ln 15$. To solve this equation, we need to understand the properties of logarithms and exponents. The equation involves the natural logarithm (ln) and the exponential function (e). The natural logarithm is the inverse of the exponential function, and it is denoted by ln.

Simplifying the Equation

To simplify the equation, we can start by using the property of logarithms that states $\ln e^x = x$. Applying this property to the given equation, we get:

$$2 \ln e^{\ln 5 x} = 2 \ln 15$$

$$2 \ln 5 x = 2 \ln 15$$

Using Logarithmic Properties

Now, we can use the property of logarithms that states $\ln a^b = b \ln a$. Applying this property to the equation, we get:

$$2 \ln 5 x = 2 \ln 15$$

$$\ln 5 x = \ln 15$$

Equating the Arguments

Since the logarithmic functions are equal, we can equate their arguments. This gives us:

$$5 x = 15$$

Solving for x

Now, we can solve for x by dividing both sides of the equation by 5:

$$x = \frac{15}{5}$$

$$x = 3$$

Conclusion

In conclusion, the true solution to the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$ is $x = 3$. This solution is obtained by simplifying the equation using logarithmic properties and equating the arguments of the logarithmic functions.

Final Answer

The final answer is $x = 3$.

Discussion

The given equation is a classic example of how logarithmic properties can be used to simplify and solve equations. The equation involves the natural logarithm and the exponential function, and it requires a deep understanding of these concepts. The solution to the equation is obtained by using the properties of logarithms and equating the arguments of the logarithmic functions.

Related Topics

• Logarithmic properties
• Exponential functions
• Natural logarithm
• Equations involving logarithms

References

• [1] "Logarithmic Properties" by Math Open Reference
• [2] "Exponential Functions" by Khan Academy
• [3] "Natural Logarithm" by Wolfram MathWorld

Additional Resources

• [1] "Logarithmic Equations" by Purplemath
• [2] "Exponential and Logarithmic Equations" by Mathway
• [3] "Solving Logarithmic Equations" by IXL
  

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Introduction

In our previous article, we explored the true solution to the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$. We simplified the equation using logarithmic properties and equated the arguments of the logarithmic functions to obtain the solution $x = 3$. In this article, we will answer some frequently asked questions related to the equation and its solution.

Q&A

Q: What is the main concept used to solve the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$?

A: The main concept used to solve the equation is the property of logarithms that states $\ln e^x = x$. This property allows us to simplify the equation and equate the arguments of the logarithmic functions.

Q: How do we simplify the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$?

A: We simplify the equation by using the property of logarithms that states $\ln e^x = x$. This gives us $2 \ln 5 x = 2 \ln 15$.

Q: What is the next step in solving the equation $2 \ln 5 x = 2 \ln 15$?

A: The next step is to use the property of logarithms that states $\ln a^b = b \ln a$. This gives us $\ln 5 x = \ln 15$.

Q: How do we equate the arguments of the logarithmic functions in the equation $\ln 5 x = \ln 15$?

A: We equate the arguments of the logarithmic functions by setting $5 x = 15$.

Q: What is the final step in solving the equation $5 x = 15$?

A: The final step is to solve for x by dividing both sides of the equation by 5. This gives us $x = \frac{15}{5}$, which simplifies to $x = 3$.

Q: What is the significance of the solution $x = 3$?

A: The solution $x = 3$ is significant because it represents the value of x that satisfies the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$.

Q: Can we use other methods to solve the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$?

A: Yes, we can use other methods to solve the equation, such as using the change of base formula or the properties of exponents. However, the method used in this article is the most straightforward and efficient way to solve the equation.

Q: What are some common mistakes to avoid when solving the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$?

A: Some common mistakes to avoid when solving the equation include:

• Not using the property of logarithms that states $\ln e^x = x$
• Not equating the arguments of the logarithmic functions
• Not solving for x correctly
• Not checking the solution for validity

Conclusion

In conclusion, the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$ can be solved using the properties of logarithms and exponents. The solution $x = 3$ is obtained by simplifying the equation and equating the arguments of the logarithmic functions. We hope that this Q&A article has provided a clear and concise explanation of the solution to the equation.

Final Answer

The final answer is $x = 3$.

Discussion

The equation $2 \ln e^{\ln 5 x} = 2 \ln 15$ is a classic example of how logarithmic properties can be used to simplify and solve equations. The solution to the equation is obtained by using the properties of logarithms and equating the arguments of the logarithmic functions. We hope that this Q&A article has provided a clear and concise explanation of the solution to the equation.

Related Topics

• Logarithmic properties
• Exponential functions
• Natural logarithm
• Equations involving logarithms

References

• [1] "Logarithmic Properties" by Math Open Reference
• [2] "Exponential Functions" by Khan Academy
• [3] "Natural Logarithm" by Wolfram MathWorld

Additional Resources

• [1] "Logarithmic Equations" by Purplemath
• [2] "Exponential and Logarithmic Equations" by Mathway
• [3] "Solving Logarithmic Equations" by IXL