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

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Solving the Equation: Uncovering the True Solution

In mathematics, equations often require a deep understanding of various concepts and techniques to solve them. One such equation is $2 \ln e^{\ln 5 x} = 2 \ln 15$. This equation involves logarithmic functions and can be solved using properties of logarithms. In this article, we will delve into the solution of this equation and explore the true solution among the given options.

The given equation is $2 \ln e^{\ln 5 x} = 2 \ln 15$. To solve this equation, we need to understand the properties of logarithmic functions. The logarithmic function $\ln x$ is the inverse of the exponential function $e^x$. This means that $\ln e^x = x$ and $e^{\ln x} = x$.

Using these properties, we can simplify the given equation. We start by applying the property $\ln e^x = x$ to the left-hand side of the equation:

2ln⁡eln⁡5x=2ln⁡5x2 \ln e^{\ln 5 x} = 2 \ln 5 x

Now, we can simplify the right-hand side of the equation using the property $e^{\ln x} = x$:

2ln⁡5x=2ln⁡152 \ln 5 x = 2 \ln 15

To simplify the equation further, we can use the property of logarithms that states $\ln a^b = b \ln a$. Applying this property to the left-hand side of the equation, we get:

2ln⁡5x=ln⁡1522 \ln 5 x = \ln 15^2

Now, we can simplify the right-hand side of the equation using the property $\ln a^b = b \ln a$:

ln⁡152=2ln⁡15\ln 15^2 = 2 \ln 15

Now that we have simplified both sides of the equation, we can equate the expressions:

2ln⁡5x=2ln⁡152 \ln 5 x = 2 \ln 15

To solve for $x$, we can divide both sides of the equation by 2:

ln⁡5x=ln⁡15\ln 5 x = \ln 15

Now that we have simplified the equation, we can solve for $x$. We start by applying the property $\ln a = \ln b$ implies $a = b$ to the equation:

5x=155 x = 15

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

x=155x = \frac{15}{5}

Now that we have solved for $x$, we can calculate the value of $x$:

x=155=3x = \frac{15}{5} = 3

In this article, we have solved the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$ using properties of logarithmic functions. We have simplified the equation, equated the expressions, and solved for $x$. The true solution to the equation is $x = 3$.

Now that we have solved the equation, we can compare our solution with the given options:

  • A. $x = 0$
  • B. $x = 3$
  • C. $x = 9$
  • D. $x = 15$

Our solution $x = 3$ matches with option B.

The final answer to the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$ is $x = 3$.
Frequently Asked Questions: Understanding the Solution to the Equation

In our previous article, we solved the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$ using properties of logarithmic functions. We simplified the equation, equated the expressions, and solved for $x$. In this article, we will address some frequently asked questions related to the solution of the equation.

A: The property of logarithms used in the solution is $\ln e^x = x$ and $e^{\ln x} = x$. These properties are essential in simplifying the given equation.

A: We simplified the left-hand side of the equation by applying the property $\ln e^x = x$. This property states that the natural logarithm of the exponential function of a number is equal to the number itself.

A: Equating the expressions is a crucial step in solving the equation. By equating the expressions, we can eliminate the logarithmic functions and solve for $x$.

A: We solved for $x$ by applying the property $\ln a = \ln b$ implies $a = b$. This property states that if the natural logarithm of two numbers are equal, then the numbers themselves are equal.

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

A: Option B is the correct answer because it matches with the solution we obtained by solving the equation. Our solution $x = 3$ is the same as option B.

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

  • Not applying the properties of logarithms correctly
  • Not simplifying the equation properly
  • Not equating the expressions correctly
  • Not solving for $x$ correctly

In this article, we have addressed some frequently asked questions related to the solution of the equation $2 \ln e^{\ln 5 x} = 2 \ln 15$. We have explained the properties of logarithms used in the solution, simplified the equation, equated the expressions, and solved for $x$. We have also provided some tips on how to avoid common mistakes when solving logarithmic equations.

For more information on logarithmic equations and properties of logarithms, please refer to the following resources:

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

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