Problem 33 - Calculus and its applications

Question

Given \(\ln 4=1.3863\) and \(\ln 5=1.6094,\) find each value. Do not use a calculator. $$ \ln \frac{1}{5} $$

Step-by-Step Solution

Answer

\(\ln \frac{1}{5} = -1.6094\)

Step 1: Understand the Logarithmic Property

We need to calculate \(\ln \frac{1}{5}\). The property of logarithms \(\ln \frac{1}{a} = -\ln a\) will be helpful here. This property tells us that the logarithm of the reciprocal of a number is equal to the negative logarithm of the number itself.

Step 2: Apply the Property to the Given Values

Using the property \(\ln \frac{1}{a} = -\ln a\), we can write \(\ln \frac{1}{5} = -\ln 5\). Now we substitute the known value of \(\ln 5\), which is 1.6094.

Step 3: Calculate the Result

Substitute \(\ln 5 = 1.6094\) into the equation \(\ln \frac{1}{5} = -\ln 5\), giving us \(\ln \frac{1}{5} = -1.6094\).

Key Concepts

Logarithm Properties

Logarithmic properties provide fundamental rules to help simplify and manipulate logarithmic expressions easily. These properties include several key rules:

• Product Rule: This states that the logarithm of a product is the sum of the logarithms of the individual factors. Mathematically, this is expressed as \( \log_b(x \, y) = \log_b x + \log_b y \).
• Quotient Rule: This property states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
• Power Rule: The logarithm of a number raised to a power is the power multiplied by the logarithm of the number: \( \log_b(x^n) = n \cdot \log_b x \).
• Change of Base Formula: This is used to change the base of a logarithm to another base: \( \log_b x = \frac{\log_k x}{\log_k b} \).

Understanding these rules allows us to rearrange and simplify logarithmic expressions within complex problems. In our problem, we used the reciprocal property: \( \ln \frac{1}{a} = -\ln a \). This helps in calculating the natural logarithm of the reciprocal of a number without a calculator.

Natural Logarithms

Natural logarithms are a specific type of logarithm whose base is the irrational number \( e \), approximately equal to 2.71828. The notation for natural logarithms is \( \ln \). They are often used in mathematics because of their natural occurrence in growth and decay processes such as population growth, radioactive decay, and continuously compounded interest.
Natural logarithms share the same properties as general logarithms, such as the product, quotient, and power rules. This allows us to use them in various mathematical manipulations like simplifying expressions or solving equations. In the exercise presented, we dealt with natural logarithms by applying the reciprocal property, showing that:

• \( \ln \frac{1}{5} = -\ln 5 \)

This property makes it easier to compute logarithms of fractions and forms a foundation in understanding more complex logarithmic applications.

Reciprocal Logarithms

The concept of reciprocal logarithms stems from understanding the relationship between a number and its reciprocal. The reciprocal of a number \( a \) is \( \frac{1}{a} \). There's a simple yet powerful logarithmic property that ties these together: \( \ln \frac{1}{a} = -\ln a \).
This property is particularly useful when we have the logarithm of a number but need to find the logarithm of its reciprocal.

• For example, given \( \ln 5 = 1.6094 \), to find \( \ln \frac{1}{5} \), you apply the property and find \( \ln \frac{1}{5} = -1.6094 \).

Understanding reciprocal logarithms also plays a role in more advanced mathematics, where expressions often involve inverses or reciprocals. Recognizing and correctly applying this property is key to efficiently solving logarithmic problems.