Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln x=-3$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\ln x = -3\) is \(x = e^{-3}\), which is approximately \(x = 0.05\) when expressed as a decimal.

Step 1: Transform the Logarithmic Equation to Exponential Form

To remove logarithm and solve for \(x\), both sides of the equation should be exponentiated using the base \(e\) as the logarithm in the given equation is natural logarithm. This yields \(e^{\ln x} = e^{-3}\). Remember, this is possible because the natural logarithm is base \(e\) and \(e\) raised to the power of natural logarithm of \(x\) is just \(x\).

Step 2: Simplify the Equation

The left hand side simplifies to \(x\) because any number raised to the logarithm of a number \(x\) using the same base gives \(x\). If we simplify the right hand side, we get \(e^{-3}\). Thus, the simplified equation now is: \(x = e^{-3}\).

Step 3: Evaluate the Expression \(e^{-3}\)

The number \(e\) (approximately 2.71828) to the power of -3 can be evaluated using a calculator to get an approximate decimal value, that is, \(x \approx 0.05\).

Step 4: Check if the Solution is in the Domain of Original Logarithmic Expression

The solution \(x = 0.05\) indeed satisfies the condition that \(x > 0\), so it is in the domain of the original logarithmic expression \(\ln x\). Therefore, the solution is valid.

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a specific logarithmic function where the base is the mathematical constant \( e \). The value of \( e \) is approximately 2.71828, and it is an irrational number that arises naturally in various branches of mathematics, especially in calculus. The natural logarithm of a number \( x \) is the power to which \( e \) must be raised in order to get \( x \). For example, \( \ln(e) = 1 \) because \( e^1 = e \), and \( \ln(1) = 0 \) because \( e^0 = 1 \).

Natural logarithms are widely used in mathematical modeling of real-world processes such as population growth, radioactive decay, and financial calculations involving compound interest. The natural logarithm has a unique property in calculus that makes differentiating exponential functions straightforward, and it forms a foundation for continuous compound interest calculations. Remember that natural logarithms only operate on positive numbers due to their definition in terms of exponential growth, adhering to the domain constraint that \( x > 0 \).

When solving equations involving natural logarithms, it's essential to understand their properties and behavior. Manipulating these equations often involves converting the natural logarithmic expressions to exponential form to simplify and solve them effectively.

Exponential Form

Transforming a logarithmic equation into its exponential form can make it easier to solve. Exponential form focuses on expressing the equation as a power of its base, which becomes particularly useful in solving logarithmic equations like \( \ln x = -3 \). When expressed in exponential form, this equation becomes \( e^{-3} \).

To convert a natural logarithmic equation \( \ln x = y \) to its exponential form, use the relationship: \( e^y = x \). By doing this, we're leveraging the property of logarithms where any number raised to the logarithm of that number results in the original value. This transformation cleans up the equation, making it linear in nature and thus far easier to solve.

Visualizing the conversion of logarithmic to exponential form might help. Imagine you need to solve \( b^y = x \) for \( y \) when you have \( \log_b(x) = y \). Here, \( e \) is our base, emphasizing the switch from logarithmic to exponential setup. The exponential form is incredibly significant as it provides a direct approach to evaluating expressions like \( e^{-3} \), which can then be approximated using a calculator for practical applications.

Domain of Logarithmic Expression

The domain of a logarithmic expression is crucial to determine whether a solution is valid. For any logarithmic function \( \ln(x) \), the domain comprises all positive real numbers. In simpler terms, the input \( x \) must be greater than zero for the natural logarithm of \( x \) to exist.

This restriction originates from the fact that you cannot take the logarithm of a non-positive number, including zero, in real number systems because no real number exists such that \( e^y = 0 \) or \( e^y \) equals a negative value. Therefore, when solving an equation like \( \ln(x) = -3 \), ensuring that the resultant \( x \) remains within its domain (\( x > 0 \)) is essential.

To verify the domain, reevaluate the solution: after solving \( x = e^{-3} \), confirm \( x > 0 \). Indeed, \( e^{-3} \) is a positive number because \( e \) raised to any real number remains positive. Checking the domain not only upholds the fundamental properties of logarithmic functions but also confirms the mathematical soundness and applicability of the solution in the given context. By ensuring your solutions lie within the domain, you avoid errors and maintain the integrity of results derived from logarithmic equations.