Problem 36 - Precalculus : Building Concepts and Connections

Question

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln x=1$$

Step-by-Step Solution

Answer

The solution to the equation \(\ln x = 1\) is \(x = e\).

Step 1: Rewrite in Exponential Form

Begin by rewriting the logarithmic equation in exponential form. The base of a natural logarithm is the constant \(e\). So the equation \(\ln x = 1\) becomes \(e^1 = x\).

Step 2: Solve for x

With this equation \(e^1 = x\), solving for \(x\) becomes straightforward. The number \(e\) raised to the power of \(1\) is just \(e\), so therefore \(x = e\).

Step 3: Check for Extraneous Solutions

Any solutions should be checked to ensure they do not violate the domain of the logarithmic function. In this case, since \(x = e\) and \(e > 0\), the solution is valid and there are no extraneous solutions.

Key Concepts

Exponential Form

When you encounter a logarithmic equation, such as \( \ln x = 1 \), a useful method for solving it is to rewrite it in exponential form. This approach stems from the basic properties of logarithms and exponents. The natural logarithm, denoted as \( \ln \), always has a base of \( e \), where \( e \approx 2.71828 \). Therefore, you can express \( \ln x = 1 \) exponentially as \( e^1 = x \). Essentially, this means "\( x \) is the number you get when you raise \( e \) to the power of 1".
By converting a logarithmic equation to exponential form, you make it easier to solve as it becomes a straightforward calculation. In this case, because \( e^1 \) simplifies directly to \( e \), we find \( x = e \). Using exponential form not only simplifies the equation but gives you a direct path to finding the solution.

Natural Logarithm

The natural logarithm, often written as \( \ln \), is a logarithm that uses the constant \( e \) as its base. The constant \( e \) is a crucial number in mathematics, specifically in calculus and complex systems. The natural log is useful for modeling growth processes like population growth or interest compounding

• \( \ln(e) = 1 \) because any base raised to the power of 1 is itself.
• Natural logs turn multiplication inside the log into addition outside: \( \ln(ab) = \ln a + \ln b \).

When you are solving logarithmic equations, recognizing the base \( e \) helps you convert the log form equations to exponential form equations, simplifying the process. In the problem \( \ln x = 1 \), you're essentially finding out how much you need to raise \( e \) to get \( x \). As the natural log of \( e \) is 1 by definition, this simplifies directly to \( x = e \). Understanding natural logarithms is essential for working with natural growth models and equations in various scientific fields.

Extraneous Solutions

Extraneous solutions are results that come from solving an equation but don't actually satisfy the original equation. They typically arise when squaring both sides of an equation, taking even roots, or introducing division by an unknown that might be zero.
In logarithmic equations, it's important to check that any solution found is within the domain of the log function. For example, \( \ln x \) is only defined for \( x > 0 \). Therefore, when you solve \( \ln x = 1 \) and find \( x = e \), you need to ensure \( e > 0 \), which is true by definition as \( e \approx 2.71828 \). This confirms the solution is not extraneous.
Thus, checking for extraneous solutions guarantees that your calculations remain meaningful and correct within the problem's constraints. This step is crucial because it prevents errors and ensures the validity of your final answers in mathematical reasoning.