Problem 32
Question
In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{1}{5} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(- \ln 5\).
1Step 1: 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: \(\ln 1 - \ln 5\).
2Step 2: 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 LogarithmProperties of LogarithmsExpressions and Simplification
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.
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 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.
- 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: \( \log_b (x^y) = y \log_b x \)
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:
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.
Other exercises in this chapter
Problem 31
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