Problem 64 - Elementary and Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \ln x=5 $$

Step-by-Step Solution

Answer

The exact solution is \(x = e^5\); approximated, \(x \approx 148.4132\).

Step 1: Understand the Equation

The equation given is \(\ln x = 5\). Here, \(\ln\) denotes the natural logarithm, which is the logarithm with base \(e\), where \(e \approx 2.71828\). We need to solve for \(x\).

Step 2: Apply the Exponential Function

To isolate \(x\), we use the property of logarithms which states that if \(\ln x = c\), then \(x = e^c\). Thus, \(x = e^5\).

Step 3: Calculate the Exact Solution

As \(x = e^5\), this is the exact solution to the given logarithmic equation.

Step 4: Approximate the Solution

Using a calculator, we find the approximate value of \(e^5\). Calculating this gives \(e^5 \approx 148.4132\).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a fundamental concept in mathematics. It operates with the base \( e \), an irrational constant approximately equal to 2.71828. This base is the backbone of continuous growth calculations in fields like biology, economics, and even physics. Unlike common logs, which are often based on 10, the natural logarithm is particularly useful in solving equations involving exponential growth or decay.
To grasp the role of \( \ln \), consider it as a function that tells us the power to which \( e \) must be raised to achieve a specific number. For instance, in the equation \( \ln x = 5 \), it's asking to find the value of \( x \) such that when \( e \) is raised to some power, it results in \( x \).
Key aspects of natural logarithms:

• It is the inverse of the exponential function with base \( e \).
• It can transform multiplicative relationships into additive ones, simplifying complex calculations.

Exponential Functions and Their Role

An exponential function is essential when you want to express growth. It's defined generally as \( f(x) = a^x \) where \( a \) is a positive constant. When this constant is \( e \), the function becomes \( e^x \), a vital tool for natural logarithms.
In the context of the equation \( \ln x = 5 \), applying exponential functions helps solve for \( x \). Because the natural logarithm and the exponential function are inverses, \( \ln x = 5 \) transforms into \( x = e^5 \). This relation is a cornerstone of solving logarithmic equations efficiently.
This duality is pivotal because:

• The exponential function helps convert logarithmic equations back to simpler form.
• It is involved in many real-life processes, including compounding interest, population growth, and radioactive decay.

Exact Solutions in Logarithmic Equations

Obtaining an exact solution means finding a precise answer, free of approximations. In logarithmic equations, exact solutions often involve calculating powers of \( e \).
For \( \ln x = 5 \), the exact solution is \( x = e^5 \). This formulation represents a precise mathematical relationship that holds true without rounding or estimation.
Exact solutions are beneficial because:

• They give us clarity and precision in mathematical results.
• They're indispensable in proofs and theoretical computations, where approximation would fall short.

While exact solutions are ideal, in practice, especially when further calculations might use these values, approximations are often necessary but should be approached carefully to maintain accuracy.

Using Approximation Techniques Effectively

Approximation techniques are invaluable when dealing with irrational numbers or complex equations, especially in practical applications. While \( e^5 \) is the exact solution, using a calculator or software can provide an approximation of this value.
For example, by calculating \( e^5 \), we get approximately 148.4132, accurate to four decimal places. Approximations like this are crucial for day-to-day computations where using an irrational number in its exact form isn't feasible.
Key things to consider when approximating:

• The desired precision level can vary, but in many cases, four decimal places are sufficient for engineering or scientific tasks.
• Being aware of the potential for small errors accumulating in further computations if initial approximations are not handled carefully.

Overall, while approximations simplify calculations, maintaining an awareness of their limitations ensures they are used appropriately and effectively.