What Is The True Solution To 2 Ln ⁡ E Ln ⁡ 5 X = 2 Ln ⁡ 15 2 \ln E^{\ln 5x} = 2 \ln 15 2 Ln E L N 5 X = 2 Ln 15 ?A. X = 3 X = 3 X = 3 B. X = 15 X = 15 X = 15 C. X = 9 X = 9 X = 9

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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 relationships between numbers. In this article, we will explore the solution to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ and determine the true value of $x$.

Understanding the Equation

The given equation is $2 \ln e^{\ln 5x} = 2 \ln 15$. To solve this equation, we need to simplify it and isolate the variable $x$. The equation involves logarithms, which can be simplified using the properties of logarithms.

Properties of Logarithms

Logarithms have several properties that can be used to simplify equations. Some of the key properties of logarithms are:

• Product Rule: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Rule: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• Power Rule: $\log_b x^y = y \log_b x$
• Change of Base Formula: $\log_b x = \frac{\log_a x}{\log_a b}$

Simplifying the Equation

Using the properties of logarithms, we can simplify the given equation as follows:

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

Using the Power Rule, we can rewrite the equation as:

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

Using the Product Rule, we can rewrite the equation as:

$\ln (e^{\ln 5x}) + \ln (e^{\ln 5x}) = \ln 15 + \ln 15$

Using the Power Rule again, we can rewrite the equation as:

$\ln (e^{\ln 5x}) = \ln 15$

Using the Change of Base Formula, we can rewrite the equation as:

$\ln (e^{\ln 5x}) = \ln 15$

Using the Power Rule again, we can rewrite the equation as:

$\ln 5x = \ln 15$

Solving for $x$

Now that we have simplified the equation, we can solve for $x$. To do this, we need to isolate the variable $x$.

Using the Change of Base Formula

Using the Change of Base Formula, we can rewrite the equation as:

$\ln 5x = \ln 15$

Using the Change of Base Formula again, we can rewrite the equation as:

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

Simplifying the equation, we get:

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

Using the Power Rule

Using the Power Rule, we can rewrite the equation as:

$\ln 5x = \ln 15$

Using the Power Rule again, we can rewrite the equation as:

$5x = 15$

Simplifying the equation, we get:

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

$x = 3$

Conclusion

In this article, we explored the solution to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$. We simplified the equation using the properties of logarithms and isolated the variable $x$. We found that the true solution to the equation is $x = 3$.

Discussion

The solution to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ is a classic example of how logarithms can be used to solve equations. The properties of logarithms, such as the product rule, quotient rule, power rule, and change of base formula, are essential tools for simplifying equations and isolating variables.

Common Mistakes

One common mistake that students make when solving equations involving logarithms is to forget to use the properties of logarithms. For example, they may try to simplify the equation by canceling out the logarithms, without using the properties of logarithms.

Tips for Solving Equations Involving Logarithms

When solving equations involving logarithms, it is essential to use the properties of logarithms to simplify the equation. Here are some tips for solving equations involving logarithms:

• Use the product rule to simplify the equation by combining the logarithms.
• Use the quotient rule to simplify the equation by subtracting the logarithms.
• Use the power rule to simplify the equation by raising the logarithm to a power.
• Use the change of base formula to simplify the equation by changing the base of the logarithm.

By following these tips and using the properties of logarithms, you can simplify equations involving logarithms and isolate the variable.

Final Answer

The final answer to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ is:

$x = 3$

This is the true solution to the equation, and it can be verified by plugging the value of $x$ back into the original equation.

References

• Logarithm Properties: A comprehensive guide to the properties of logarithms, including the product rule, quotient rule, power rule, and change of base formula.
• Solving Equations Involving Logarithms: A step-by-step guide to solving equations involving logarithms, including tips and tricks for simplifying the equation.
• Mathematics: A comprehensive guide to mathematics, including algebra, geometry, calculus, and more.

Note: The final answer is A. $x = 3$.

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Introduction

In our previous article, we explored the solution to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$. We simplified the equation using the properties of logarithms and isolated the variable $x$. In this article, we will answer some of the most frequently asked questions about solving the equation $2 \ln e^{\ln 5x} = 2 \ln 15$.

Q&A

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

A: The main concept behind solving the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ is the use of logarithmic properties, specifically the product rule, quotient rule, power rule, and change of base formula.

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

A: To simplify the equation $2 \ln e^{\ln 5x} = 2 \ln 15$, you can use the properties of logarithms. Specifically, you can use the product rule to combine the logarithms, the quotient rule to subtract the logarithms, the power rule to raise the logarithm to a power, and the change of base formula to change the base of the logarithm.

Q: What is the final answer to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$?

A: The final answer to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ is $x = 3$.

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

A: Yes, you can use other methods to solve the equation $2 \ln e^{\ln 5x} = 2 \ln 15$. For example, you can use the exponential function to rewrite the equation and then solve for $x$.

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

A: Some common mistakes to avoid when solving the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ include:

• Forgetting to use the properties of logarithms
• Canceling out the logarithms without using the properties of logarithms
• Not isolating the variable $x$

Q: How can I verify the solution to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$?

A: To verify the solution to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$, you can plug the value of $x$ back into the original equation and check if it is true.

Conclusion

In this article, we answered some of the most frequently asked questions about solving the equation $2 \ln e^{\ln 5x} = 2 \ln 15$. We covered topics such as the main concept behind solving the equation, simplifying the equation, the final answer, alternative methods, common mistakes, and verifying the solution.

Discussion

Solving the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ requires a good understanding of logarithmic properties and how to apply them to simplify the equation. By following the steps outlined in this article, you can solve the equation and verify the solution.

Tips for Solving Equations Involving Logarithms

When solving equations involving logarithms, it is essential to use the properties of logarithms to simplify the equation. Here are some tips for solving equations involving logarithms:

• Use the product rule to simplify the equation by combining the logarithms.
• Use the quotient rule to simplify the equation by subtracting the logarithms.
• Use the power rule to simplify the equation by raising the logarithm to a power.
• Use the change of base formula to simplify the equation by changing the base of the logarithm.

By following these tips and using the properties of logarithms, you can simplify equations involving logarithms and isolate the variable.

Final Answer

The final answer to the equation $2 \ln e^{\ln 5x} = 2 \ln 15$ is:

$x = 3$

This is the true solution to the equation, and it can be verified by plugging the value of $x$ back into the original equation.

References

• Logarithm Properties: A comprehensive guide to the properties of logarithms, including the product rule, quotient rule, power rule, and change of base formula.
• Solving Equations Involving Logarithms: A step-by-step guide to solving equations involving logarithms, including tips and tricks for simplifying the equation.
• Mathematics: A comprehensive guide to mathematics, including algebra, geometry, calculus, and more.