Problem 37 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln x=-2.3 $$

Step-by-Step Solution

Answer

Exact solution: \( x = e^{-2.3} \), Approximation: \( x \approx 0.1003 \).

Step 1: Understand the Equation

We need to solve the equation \( \ln x = -2.3 \). The \( \ln \) function (natural logarithm) is the inverse of the exponential function with base \( e \).

Step 2: Use Inverse Operations

To solve the equation \( \ln x = -2.3 \), we use the property that if \( \ln x = a \), then \( x = e^a \). Therefore, we have \( x = e^{-2.3} \).

Step 3: Calculate the Exact Solution

Calculating \( e^{-2.3} \) gives us the exact solution \( x = e^{-2.3} \). This describes \( x \) in terms of the mathematical constant \( e \).

Step 4: Compute the Approximation

Using a calculator, evaluate \( e^{-2.3} \). This gives us approximately \( x \approx 0.1003 \) when rounded to four decimal places.

Key Concepts

Exponential Function

The exponential function is one of the most important concepts in mathematics. It is often represented as \( f(x) = e^x \), where \( e \) is the base of the natural logarithm. The function describes how quantities grow at a consistent relative rate, making it crucial in fields like finance, physics, and biology.

The exponential function has some key properties:

• The function is always positive for all real numbers \( x \).
• The graph of \( e^x \) is an increasing curve that starts above zero and rises sharply.
• It passes through the point \( (0,1) \) since \( e^0 = 1 \).

In simple terms, the exponential function transforms multiplication into addition, which is why it's deeply intertwined with the concept of logarithms. When we solve an equation like \( \, e^{-2.3} \, \), we are using a specific outcome of the exponential function.

Inverse Operations

Inverse operations are operations that "undo" each other. In the context of logarithms and exponentials, they are particularly essential.

When we say that logarithms and exponential functions are inverses, it means that the logarithm function \( \ln x \) will undo an exponential function with base \( e \), and vice versa. This is captured by the equation: \[ \ln(e^x) = x \text{ and } e^{\ln x} = x \]

For example, solving \( \ln x = -2.3 \) requires translating this using the inverse exponential operation. Therefore, applying the inverse operation to both sides leads us to \( x = e^{-2.3} \). This enables us to "reverse" the natural logarithm function to find the exact value of \( x \).

Understanding inverse operations is crucial to solving equations involving exponential and logarithmic expressions.

Mathematical Constant

The mathematical constant \( e \), approximately 2.71828, is fundamental to exponential and logarithmic functions. Much like the famous \( \pi \), \( e \) is an irrational number, meaning it cannot be written as a simple fraction, and its decimal representation goes on forever without repeating.

This constant emerges naturally in many areas of mathematics:

• In calculus, as the base of natural logarithms and the limit of continuous compound interest.


• In complex analysis, where \( e^{i\pi} + 1 = 0 \), known as Euler's identity, elegantly ties together fundamental constants \( e, i, \pi \).

The use of \( e \) in calculations, as seen in \( x = e^{-2.3} \), highlights its importance in expressing and working with growth processes and decay patterns.

Approximations

In mathematics, approximations are used when exact values are complex or impossible to derive in a straightforward manner. Approximations provide a way to work practically with numbers that have more precision than needed.

When you calculate \( e^{-2.3} \), it does not yield a simple rational number. Instead, the calculator shows several decimal places, such as \( 0.1002588...\). However, for practical purposes, we often round these numbers to make calculations manageable, conveying this solution as \( x \approx 0.1003 \) rounded to four decimal places.

This kind of rounding can be particularly useful in science and engineering where numbers need to be concise, yet precise enough for usage in real-world scenarios. Understanding the balance between exactness and practicality is a valuable skill in both academic and professional contexts.