Problem 38 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln x=-3.7 $$

Step-by-Step Solution

Answer

Exact solution: \( x = e^{-3.7} \), Approximation: 0.0249

Step 1: Understand the equation

The given equation is \( \ln x = -3.7 \). This means we need to find the value of \( x \) such that its natural logarithm equals \(-3.7\).

Step 2: Remember the properties of logarithms

Recall that the natural logarithm \( \ln x \) is the power to which the base \( e \) must be raised to get \( x \). Thus, if \( \ln x = -3.7 \), then \( x = e^{-3.7} \).

Step 3: Calculate the exact solution

Since the equation states \( x = e^{-3.7} \), the exact solution can be expressed as \( x = e^{-3.7} \).

Step 4: Calculate a decimal approximation

Use a calculator to find the value of \( e^{-3.7} \). The approximate value, rounded to four decimal places, is 0.0249.

Key Concepts

Natural Logarithm

A natural logarithm is a specific type of logarithm that has the base \( e \). The constant \( e \) is an irrational number approximately equal to 2.71828. When we say \( \ln x \), we mean the power to which \( e \) must be raised to result in \( x \). For instance, if \( \ln x = y \), then \( e^y = x \). This relationship ties the exponential function closely with the natural logarithm.
In this exercise, \( \ln x = -3.7 \), indicating that we need to find the number \( x \) such that \( e \) raised to the power of \(-3.7\) equals \( x \). Understanding this helps us determine that the reverse operation involves exponentiation with \( e \).
This is a fundamental concept in solving logarithmic equations and is a backbone for understanding exponents and logarithms.

Properties of Logarithms

Logarithms, including natural logarithms, have certain properties that are frequently used to simplify calculations and solve equations. One key property is the inverse relationship between exponentiation and logarithms. This is crucial for translating between the logarithmic form and its equivalent exponential form, as seen in the equation \( \ln x = -3.7 \).
Some important properties of logarithms are:

• The product rule: \( \ln(ab) = \ln a + \ln b \)
• The quotient rule: \( \ln \frac{a}{b} = \ln a - \ln b \)
• The power rule: \( \ln(a^b) = b \ln a \)
• The base change rule: \( \ln a = \frac{\log_b a}{\log_b e} \)

The application of these properties aids in transforming and simplifying complex logarithmic expressions. For the given problem, recognizing that \( x = e^{-3.7} \) directly comes from exploiting the inverse relationship of exponentials and logarithms.

Decimal Approximation

When solving logarithmic equations, obtaining an approximate decimal value can provide a clearer numerical understanding. While the exact solution is nice, practical situations often require a numerical approximation that is usable with a given degree of precision.
In our case, after finding the exact solution \( x = e^{-3.7} \), we use a calculator to approximate \( e^{-3.7} \). Calculators and computational tools typically do this efficiently because they are programmed to handle these transcendental numbers precisely. The result, rounded to four decimal places, is \( 0.0249 \).
Knowing how to obtain this approximate value is essential when graphing functions or applying results in real-world contexts.

Exact Solution

Finding the exact solution in logarithmic equations involves setting up the equation such that it can be expressed in a precise, non-decimal form. For logarithmic equations involving natural logarithms, this usually means representing the solution using the constant \( e \) and exponents.
In solving \( \ln x = -3.7 \), we rework the equation into its exponential form: \( x = e^{-3.7} \). This expression is considered exact because it fully represents the solution without rounding errors or decimal truncation. It's an exact answer that maintains accuracy.
Exact solutions are perfect for further algebraic manipulation or when precision is paramount. This representation makes exact solutions highly valuable in theoretical mathematics and complex analyses.