Solving the Equation ln x 2 ln (1 - x)

The natural logarithm, denoted as ln, often appears in various fields of mathematics and science. When dealing with more complex logarithmic equations such as ln x 2 ln (1 - x), it is essential to employ algebraic manipulation and properties of logarithms to find a solution. In this article, we will walk through the steps to solve the equation ln x 2 ln (1 - x) effectively.

Understanding the Equation

The given equation is ln x 2 ln (1 - x). Breaking it down, you can see that the natural logarithm of x is equal to twice the natural logarithm of (1 - x). Let's simplify this equation using some basic properties of logarithms.

Step 1: Applying Logarithm Properties

By using the logarithm property stated as ln a - ln b ln (a/b), we can combine the two logarithms on the left-hand side:ln x - ln (1 - x) 2This simplifies to:ln left(frac{x}{1 - x}right) 2

Step 2: Exponentiating Both Sides

To eliminate the natural logarithm, we can exponentiate both sides of the equation using the base e (since ln a b implies a e^b):left(frac{x}{1 - x}right) e^2

Step 3: Simplifying the Equation

At this point, we have the simplified equation:frac{x}{1 - x} e^2To isolate x, we can cross-multiply:1x 1 - xe^2This can be rearranged to:x xe^2 1 e^2Factorizing the left-hand side:x(1 e^2) 1 e^2Finally, dividing both sides by (1 e^2):x frac{1 e^2}{1 e^2}Thus, the solution to the equation is:x 1

Conclusion

In conclusion, we have successfully solved the equation ln x 2 ln (1 - x) through a series of steps involving logarithm properties and algebraic manipulation. It is important to understand the fundamental properties of logarithms and how to combine and isolate variables to solve such equations effectively. Solving the Equation ln x 2 ln (1 - x)

The equation ln x 2 ln (1 - x) is a fundamental problem in logarithmic mathematics. By leveraging the properties of logarithms and algebraic techniques, we can systematically solve such equations. Let's explore the solution in detail.

Understanding the Equation

The given equation is written as ln x 2 ln (1 - x).

Breaking down the left-hand side, it can be seen that this is equivalent to ln x - ln (1 - x) 2 using the logarithm property ln a - ln b ln (a/b).

This further simplifies to:

ln left( frac{x}{1 - x} right) 2

Exponentiating Both Sides

To eliminate the natural logarithm, we can exponentiate both sides using the base e:

frac{x}{1 - x} e^2

Simplifying and Solving for x

Cross-multiplying and rearranging terms, we get:

x xe^2 1 e^2

This can be factorized as:

x(1 e^2) 1 e^2

Dividing both sides by (1 e^2):

x frac{1 e^2}{1 e^2}

Simplifying the final expression, we obtain the solution:

x 1

Conclusion

By following these structured steps, we have successfully solved the equation ln x 2 ln (1 - x).

Understanding the nature of logarithmic functions and utilizing algebraic techniques ensures that such complex problems can be tackled effectively and accurately.