Problem 41 - Calculus and its applications

Question

Given \(\ln 4=1.3863\) and \(\ln 5=1.6094,\) find each value. Do not use a calculator. $$ \ln \left(\frac{4}{e}\right) $$

Step-by-Step Solution

Answer

\( \ln\left(\frac{4}{e}\right) = 0.3863 \)

Step 1: Recognize the Logarithmic Property

We start by using the property of logarithms that states: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \). We will apply this to \( \ln\left(\frac{4}{e}\right) \).

Step 2: Apply the Logarithmic Property

Substitute \( a = 4 \) and \( b = e \) into the property: \( \ln\left(\frac{4}{e}\right) = \ln 4 - \ln e \).

Step 3: Use Known Values

Use the given values \( \ln 4 = 1.3863 \) and the fact that \( \ln e = 1 \). Substitute these into the expression: \( \ln\left(\frac{4}{e}\right) = 1.3863 - 1 \).

Step 4: Simplify

Calculate the expression: \( \ln\left(\frac{4}{e}\right) = 0.3863 \).

Key Concepts

Understanding Natural Logarithm

A natural logarithm is denoted by ln and is a special kind of logarithm that has the base of Euler's number, commonly known as "e." This constant, approximately equal to 2.71828, is irrational and is the foundation of many mathematical concepts, especially in calculus and complex analysis.
When you see ln followed by a number or an expression, it asks the question: "To what power must e be raised to get this number?" This is analogous to asking how many times a base number (e) must be multiplied to achieve a particular value.
For example:

• If you encounter ln(1), the answer is 0 because raising e to the power of 0 results in 1.
• ln(e) is equal to 1 because raising e to its first power gives back e.

The natural logarithm is crucial for growth problems, decay processes, and various logarithmic identities.

Logarithm Simplification Techniques

Simplifying logarithmic expressions is an essential skill in algebra. The goal is to use properties of logarithms to rewrite complex expressions into simpler or more manageable forms. This often involves the combination or separation of logs.
Consider some commonly used techniques:

• Product Property: This property states that the logarithm of a product is the sum of the logarithms of the factors: \[ \ln(ab) = \ln a + \ln b \]
• Quotient Property: For the logarithm of a quotient, it subtracts the logarithm of the denominator from the logarithm of the numerator, as shown: \[ \ln\left( \frac{a}{b} \right) = \ln a - \ln b \]
• Power Property: This one states that the logarithm of a power is the exponent times the logarithm of the base: \[ \ln(a^b) = b \cdot \ln a \]

These properties empower you to transform complicated logarithmic expressions into ones that are easier to handle, as demonstrated in the original solution.

Exploring Properties of Logarithms

Properties of logarithms are vital tools in mathematics that allow us to manipulate and solve logarithmic expressions effectively. They provide a framework to "deconstruct" logarithms into their components, thus simplifying calculations.
Let's look at why these properties are useful:

• The product property helps condense multiple logs into a single log expression, which is often more convenient for both solving and calculation.
• The quotient property aids in breaking down complex fractions in logarithmic form, as seen in simplifying \( \ln\left(\frac{4}{e}\right) \) to \( \ln 4 - \ln e \).
• The power property makes it easy to deal with powers and roots within logs, turning these into multiplication outside of the log function.

These properties work together to simplify expressions, solve equations, and make it much easier to manipulate logarithms in various mathematical contexts.