Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (1-x)=\frac{1}{2}$$

Step-by-Step Solution

Answer

The solution is \(x = 1 - e^{\frac{1}{2}}\).

Step 1: Express the natural logarithm equation

The given equation is \( \ln(1-x) = \frac{1}{2} \). This equation involves a natural logarithm, which can be solved by transforming it into an exponential form.

Step 2: Convert the logarithmic equation to exponential form

Recall that if \( \ln(a) = b \), then \( a = e^b \). Applying this to our equation, we have:\[1-x = e^{\frac{1}{2}}\]So, we express \(1-x\) in terms of an exponential by raising \(e\) to the power of \(\frac{1}{2}\).

Step 3: Solve for \(x\)

To find \(x\), solve the equation:\[1-x = e^{\frac{1}{2}}\]Rearrange to isolate \(x\):\[x = 1 - e^{\frac{1}{2}}\]

Step 4: Verify the solution with a calculator

Calculate \( e^{\frac{1}{2}} \) using a calculator to ensure accuracy. The approximate value of \( e^{\frac{1}{2}} \) is about 1.6487. Substitute this back into the expression for \(x\) to confirm:\[x \approx 1 - 1.6487 \approx -0.6487\]This verification step ensures the solution is consistent with the initial equation.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a logarithm with base \( e \), where \( e \) is the mathematical constant approximately equal to 2.71828. The natural logarithm is the inverse operation of raising \( e \) to a power. In other words, it tells us the power to which \( e \) must be raised to obtain a given number.

• Natural logarithms are particularly useful in solving problems involving growth and decay models in science, finance, and other fields.
• When working with natural logarithms, it is important to remember the properties of logarithms, such as the product, quotient, and power rules.

For the equation \( \ln(1-x) = \frac{1}{2} \), we know that \( \ln(1-x) \) represents the power that \( e \) must be raised to in order to obtain \( 1-x \). This provides a foundation for converting this into an exponential equation, which will help us solve for \( x \). Understanding this fundamental relationship is key to solving logarithmic equations effectively.

Exponential Form

Converting logarithmic equations into their exponential form is a crucial step in making them easier to solve. The central principle here is:

• If \( \ln(a) = b \), then this is equivalent to saying \( a = e^b \).

This is because the natural logarithm is the inverse function of the exponential function with base \( e \).
In the provided exercise, the equation \( \ln(1-x) = \frac{1}{2} \) becomes \( 1-x = e^{\frac{1}{2}} \) when expressed in exponential form. This transformation allows us to work directly with \( e \) raised to a power, making it easier to isolate and solve for \( x \).

• Expressing equations in exponential form can demystify logarithms and lead to straightforward solutions.
• It is a common technique used in many mathematical problems to switch between logarithmic and exponential forms.

Recognizing this conversion makes it easier to navigate through logarithmic problems and enhances problem-solving skills in mathematics.

Solving Equations

Once the logarithmic equation is converted to its exponential form, solving it becomes a matter of straightforward algebraic manipulation. Our transformed equation, \( 1-x = e^{\frac{1}{2}} \), is a simple linear equation that can be solved for \( x \) with ease.
To isolate \( x \), you perform basic algebraic operations:

• Subtract \( e^{\frac{1}{2}} \) from 1 to solve for \( x \):

\[ x = 1 - e^{\frac{1}{2}} \]This calculation gives us the exact value of \( x \).
It is always good to verify your solution by plugging it back into the original equation to see if it holds true. Also, using a calculator to check the decimal approximation, like calculating \( e^{\frac{1}{2}} \approx 1.6487 \), ensures that we have made no mistakes during the process.

• Verification is important for confirming the accuracy of your solution.
• This step distinguishes approximate results from exact answers, which is crucial in mathematical problem-solving.

By following these steps, students can efficiently solve logarithmic equations and develop a deeper understanding of the relationship between logarithmic and exponential functions.