Problem 65 - Elementary and Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=-3.71 $$

Step-by-Step Solution

Answer

The solution is approximately 0.0246.

Step 1: Understand the Equation

We start with the equation \( \ln x = -3.71 \). The natural logarithm function \( \ln \) tells us that \( x \) is represented in the form of \( e^y = x \) where \( y = \ln x \). Here, \( y \) is given as \(-3.71\). Our goal is to find \( x \).

Step 2: Exponentiate Both Sides

To solve for \( x \), we need to eliminate the natural logarithm by exponentiating both sides of the equation. This means we will take \( e \), the base of natural logarithms, and raise it to the power of both sides:\[ e^{\ln x} = e^{-3.71}\]

Step 3: Simplify Using Logarithmic Identity

Since the exponential and logarithmic functions are inverses, \( e^{\ln x} = x \). Therefore, the equation simplifies to:\[x = e^{-3.71}\]

Step 4: Calculate the Exponent

Next, we need to calculate \( e^{-3.71} \) using a calculator. This will give us the value of \( x \).

Step 5: Round to Four Decimal Places

Using a calculator, the value of \( e^{-3.71} \) calculates approximately to 0.024603. Rounding this to four decimal places gives us the final answer.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a mathematical function that tells you the power to which the base \( e \) (approximately 2.718) must be raised to produce the number \( x \). This logarithm is commonly used in mathematics, especially in calculus and exponential growth calculations.

• Its inverse function is the exponential function \( e^x \).
• \( \ln x \) is specifically defined for positive \( x \).
• Converts multiplication into addition, making complex equations simpler.

In our case, since we have the equation \( \ln x = -3.71 \), we are essentially given the exponent to which \( e \) should be raised to achieve \( x \). Understanding how natural logarithms work is crucial in solving and simplifying equations where logarithms are involved.

Exponentiation

Exponentiation is the process of raising a number to a power. In simpler terms, it means multiplying a number by itself a certain number of times defined by the exponent.

• It's denoted by a base number and an exponent, for example, \( t^n \).
• The number \( e \) is a special constant base, approximately equal to 2.718.
• Exponentiation is frequently used to reverse the effect of logarithms.

In the context of solving \( \ln x = -3.71 \), we exponentiate both sides to remove the logarithm. Thus, \( e^{\ln x} = e^{-3.71} \). Recognizing exponentiation helps make sense of how we isolate variables and solve equations that involve \( \ln \).

Rounding Decimal Places

Rounding decimal places is an important concept in achieving precision without excessive detail. For many mathematical and real-world applications, rounding helps in maintaining a controlled level of accuracy.

• Helps in representing numbers in a simplified format without unnecessary digits.
• Ensures consistency in the reported results, especially in studies requiring high precision.
• Four decimal places means to represent a number with four digits after the decimal point.

After calculating \( e^{-3.71} \) which approximates to 0.024603, we round it to four decimal places to obtain 0.0246. This rounding is crucial in ensuring results are both accurate and practical for reporting and further use.

Inverse Functions

Inverse functions are a core concept in mathematics, involving the reversal of the effect of a function. When two functions are inverses, applying one function following the other will return the original value.

• If \( f(x) \) is a function, then the inverse \( f^{-1}(x) \) returns the input of \( f(x) \).
• The natural logarithm \( \ln x \) and the exponential function \( e^x \) are inverses of each other.
• Using inverse functions is a powerful technique to solve equations.

In our problem, since \( \ln x \) and \( e^x \) are inverses, \( e^{\ln x} = x \). Thus, exponentiating \( \ln x \) with base \( e \) cancels out the logarithm, directly solving for \( x \). Understanding inverse functions allows us to efficiently tackle and solve complex equations including logs and exponents.