Question

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{1}{5} $$

Short answer

The simplified expression is \(- \ln 5\).

Step-by-step solution

Express the Fraction inside the Logarithm as a Difference of Logarithms

Using the property of logarithms \(\ln a/b = \ln a - \ln b\), substitute \(a =1 \) and \(b = 5\). The expression becomes: \(\ln 1 - \ln 5\).

Simplify the Difference of Logarithms

The natural logarithm of 1 (\(\ln 1\)) is 0. Therefore, the final expression is: \(0 - \ln 5\), which can be further simplified to \(- \ln 5\).

Key concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a logarithm where the base is the mathematical constant \( e \), approximately equal to 2.71828. It's used extensively in mathematics, especially in calculus, because of its unique property of having its derivative \( d / dx(\ln x) = 1/x \). This property simplifies many calculus problems and is why it's favored in continuous growth models.
Natural logarithms appear in various applications such as exponential growth and decay, compound interest, and in solving differential equations. Understanding \( \ln x \) thoroughly helps grasp the nuances of these applications.
Keep in mind that \( \ln 1 = 0 \) since \( e^0 = 1 \). This identity frequently appears in logarithmic simplifications and solutions.

Properties of Logarithms

Understanding properties of logarithms is essential to simplify complex expressions and solve logarithmic equations effectively. Here are the main rules you need:

• The Product Rule: \( \log_b (xy) = \log_b x + \log_b y \)
• The Quotient Rule: \( \log_b (a/b) = \log_b a - \log_b b \)
• The Power Rule: \( \log_b (x^y) = y \log_b x \)

In our specific example, the Quotient Rule was applied. The expression \( \ln \frac{1}{5} \) was split into \( \ln 1 - \ln 5 \) by setting \( a = 1 \) and \( b = 5 \).
The use of these rules not only simplifies expressions but also aids in solving logarithmic equations by isolating variables, thus making complex multiplication, division, and exponentiation problems much simpler.

Expressions and Simplification

Simplification of logarithmic expressions often results in more manageable and readable forms. The goal is to reframe expressions using properties of logarithms that make them easier to handle mathematically.
In the example provided, \( \ln \frac{1}{5} \) simplifies to \(-\ln 5\), using both the Quotient Rule of logarithms and the understanding that \( \ln 1 = 0 \). Breaking down the expression into these components allows easier computational steps.
Frequently, simplifying expressions involves:

• Breaking down complex fractions into sums or differences of logs.
• Reducing powers into multipliers within the log expression using the Power Rule.
• Recognizing constants or simple values like \( \ln 1 \), which directly simplify to known values such as 0.

Overall, mastering simplification leads to clarity and ease in solving more intricate mathematical problems, underlining why each step and rule is significant.