Problem 62 - Elementary and Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=2.6490 $$

Step-by-Step Solution

Answer

\(x \approx 14.1390\)

Step 1: Understand the Natural Logarithm

The given equation is \( x = 2.6490\). The natural logarithm \( x\) refers to a logarithm with base \(e\), where \(e\) is approximately 2.7183.

Step 2: Convert the Logarithmic Equation to Exponential Form

To solve the equation \( x = 2.6490\), we convert it to its exponential form. The equivalence \((x) = y \) can be rewritten as \(e^y = x\). Thus, \(e^{2.6490} = x\).

Step 3: Calculate the Exponential

Using a calculator, compute \(e^{2.6490}\). This will give the value of \(x\).

Step 4: Provide the Rounded Solution

The result from the calculator for \(e^{2.6490}\) is approximately 14.1390. Make sure to express this solution to four decimal places.

Key Concepts

Natural Logarithm

The natural logarithm is a type of logarithm with a special base denoted as 'e'. The number 'e' is an irrational constant approximately equal to 2.7183. It often appears in various areas of mathematics, especially in calculus, because of its natural properties related to growth and rates of change. When you see \( \ln x \), this represents the natural logarithm of \( x \). In essence, it is the logarithm to the base \( e \). This is different from the common logarithm which typically has a base of 10.

• The equation \( \ln x = 2.6490 \) tells us the power to which \( e \) must be raised to get \( x \).
• Since \( e \approx 2.7183 \), when you put a power to 'e', you're exploring natural exponential growth.
• This aligns with the inverse relationship between logarithms and exponents, central to solving many logarithmic equations.

When dealing with natural logarithms, you can often transition to exponential form to solve them, making computations more straightforward.

Exponential Form

Transforming a logarithmic equation into exponential form is a crucial step in solving it. This takes advantage of the relationship between logarithms and exponents, where the exponent provides insight into the value of a logarithmic expression. For the given problem:

• Start with the logarithmic equation: \( \ln x = 2.6490 \).
• Rewrite the equation in exponential form, which in this context uses the base \( e \): \( e^{2.6490} = x \).
• This transformation essentially states that \( e \) raised to the power of 2.6490 gives \( x \).

This conversion is helpful because exponential operations are usually easier to compute, especially simply using a calculator.

Logarithm Base e

Understanding logarithms with the base \( e \), or natural logs, is crucial for tackling various exponential and logarithmic equations. The base \( e \) naturally occurs in phenomena characterized by constant growth rates, such as population growth or interest calculations, which is why it's so extensively used in different scientific fields. In solving \( \ln x = 2.6490 \), the base \( e \) is key:

• Since \( \ln x \) indicates a natural logarithm, base \( e \) reflects exponential growth common in natural processes.
• It expresses calculations in a realistic and natural manner within real-world contexts like biological processes or financial calculations.
• The fact that \( e \approx 2.7183 \) makes it distinctively beneficial in continuous growth models compared to other bases like 10.

Mastering this means recognizing how logarithmic operations link to their exponential equivalents, thereby enabling easier and more effective computational results.