Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln x=-3\)

Step-by-Step Solution

Answer

The solution to the logarithmic equation \( \ln x = -3 \) approximated to three decimal places is \( x = 0.050 \).

Step 1: Write the Given Equation

The given logarithmic equation is \( \ln x = -3\).

Step 2: Apply anti-logarithm to the Equation

Taking anti-logarithm of both sides of the equation gives us \( x = e^{-3} \), where \( e \) approximates 2.71828.

Step 3: Calculate the Value of x

Using a calculator, compute the value of \( x \) by evaluating \( e^{-3} \).

Step 4: Approximating the Value

To obtain the result accurate to 3 decimal places, round your computed value of x accordingly.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a logarithm with base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It is a fundamental concept in mathematics, particularly in calculus and complex analysis. The natural logarithm is unique because of its relationship to growth processes and its ease of differentiation.

• The natural logarithm of a number \( x \) is the power to which \( e \) must be raised to obtain \( x \).
• For example, \( \ln e = 1 \), since \( e^1 = e \).
• The function \( \ln(x) \) is only defined for \( x > 0 \).

Understanding natural logarithms is crucial for solving equations involving exponential growth or decay.

Anti-logarithm

Anti-logarithms are the inverse operation of logarithms. In the context of the natural logarithm, the anti-logarithm involves the exponential function. To find the anti-logarithm of a natural logarithm like \( \ln x \), we rewrite it in terms of an exponent to solve for \( x \). Particularly:

• For a natural logarithm \( \ln x = y \), the equivalent statement using anti-logarithm is \( x = e^y \).
• This conversion allows us to solve logarithmic equations grounded in the natural logarithm.
• Anti-logarithms are pivotal in converting logarithmic expressions to exponential ones, facilitating calculations and interpretations.

This step was crucial for transforming the original equation \( \ln x = -3 \) into an accessible form \( x = e^{-3} \).

Exponential Function

The exponential function is a mathematical expression involving the constant \( e \) raised to the power of a variable or a number, denoted as \( e^x \). It models a vast array of natural phenomena, from population growth to radioactive decay.

• The exponential function with base \( e \) is denoted as \( f(x) = e^x \).
• It's important for its unique property that the rate of change of \( e^x \) is proportional to the function's value itself, i.e., the derivative of \( e^x \) is also \( e^x \).
• In solving \( \ln x = -3 \), the exponential function enabled the conversion to \( x = e^{-3} \), providing a means to find \( x \).

Understanding the exponential function helps in analyzing the solution’s behavior and deriving accurate approximations.

Approximation Methods

In mathematics, especially in solving equations analytically, approximation methods play an integral role. They help in determining solutions with practical utility. Here, evaluating \( e^{-3} \) necessitates approximation due to \( e \)'s irrational nature.

• Using a calculator, one can approximate the value of \( e^{-3} \) to the specified precision.
• This result provides an estimate of \( x \) to three decimal places: approximately 0.050.
• Approximations are critical in solutions involving roots or irrational numbers, offering a tangible way to understand and communicate results.

Approximating values ensures effective communication of mathematically derived results in understandable terms.