Problem 3
Step-by-Step Solution
Verified Answer
Using the inequality \(\ln(x)\leq x-1\), apply it to \(\frac{b}{a}\) to find \(\ln \left(\frac{b}{a}\right) \leq \frac{b-a}{a}\) and show \(\frac{b-a}{b} \leq \ln \left(\frac{b}{a}\right)\) by noting that \(\ln(1) = 0\). Combine to form the inequality chain.
1Step 1: Understand the Properties of Logarithms
Recognize that \(\ln(x)\leq x-1\) for all \(x>0\), with equality if and only if \(x=1\). This property will be used to derive the inequalities.
2Step 2: Apply the Logarithmic Property to the Right Side of the Inequality
Apply the logarithmic property to \(\frac{b}{a}\), knowing that \(\frac{b}{a} > 0\), to get \(\ln \left(\frac{b}{a}\right) \leq \frac{b}{a} - 1\). Since \(a < b\), we know \(\frac{b}{a} > 1\), making the inequality valid.
3Step 3: Simplify the Resulting Inequality
Simplify \(\frac{b}{a} - 1\) to get \(\frac{b-a}{a}\). Thus, you have \(\ln \left(\frac{b}{a}\right) \leq \frac{b-a}{a}\).
4Step 4: Apply the Logarithmic Property to the Left Side of the Inequality
Start from \(\ln \left(\frac{b}{b}\right) \leq \ln \left(\frac{b}{a}\right)\) which simplifies to \(0 \leq \ln \left(\frac{b}{a}\right)\) because \(\frac{b}{b} = 1\) and \(\ln(1) = 0\).
5Step 5: Construct the Final Inequality Chain
Combine the derived inequalities \(\frac{b-a}{b} = 1 - \frac{a}{b} \leq \ln \left(\frac{b}{a}\right) \leq \frac{b-a}{a}\). This maintains the truth of both original inequalities as \(0 < a \leq b\) ensures \(\frac{a}{b} \leq 1\) and \(\frac{b}{a} \geq 1\).
Key Concepts
Properties of LogarithmsLogarithmic InequalitiesInequality ProofsLogarithm Rules
Properties of Logarithms
Logarithms are mathematical functions that help in solving equations where the unknown is part of the exponent. The properties of logarithms are essential for understanding how to work with logarithmic expressions and inequalities.
Some of the most important properties include:
Some of the most important properties include:
- The product rule: \(\ln(xy) = \ln(x) + \ln(y)\)
- The quotient rule: \(\ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y)\)
- The power rule: \(\ln(x^k) = k\cdot\ln(x)\)
- Change of base formula: \(\ln(x) = \frac{\log_c(x)}{\log_c(e)}\), where \(c\) is any positive number.
- The logarithm of 1: \(\ln(1) = 0\)
- Logarithm of a product: \(\ln(a) + \ln(b) = \ln(ab)\)
Logarithmic Inequalities
Logarithmic inequalities, such as \(\ln(x) > a\) or \(\ln(x) < b\), involve a logarithm that is being compared by an inequality sign to another term. Solving these inequalities often requires using the properties of logarithms and understanding the behavior of the logarithmic function.
To solve a logarithmic inequality:
To solve a logarithmic inequality:
- Isolate the logarithmic expression on one side of the inequality.
- Convert the logarithmic inequality into an exponential form, as the exponential function is the inverse of the logarithmic function.
- Solve the resulting inequality, keeping in mind the domain of logarithmic functions, which is all positive real numbers.
- Check the solution for extraneous solutions that might arise from the earlier steps.
Inequality Proofs
Inequality proofs involve demonstrating that a relationship holds between two expressions for all acceptable values of the variables involved. Proofs for inequalities can include algebraic manipulation, applying known inequalities, and using direct or indirect argumentation.
When proving an inequality, it is important to:
When proving an inequality, it is important to:
- Consider the domain of the variables.
- Keep the inequality in the correct direction throughout manipulations.
- Use properties of numbers, such as the transitive property (if \(a < b\) and \(b < c\), then \(a < c\)) to build logical sequences of inequalities.
- Occasionally divide the proof process into cases if the behavior of the inequality changes due to the variable values.
Logarithm Rules
Logarithm rules are sets of guidelines that dictate how to manipulate logarithmic expressions. These include not just the properties of logarithms, but also how logarithms interact with inequalities.
Specific logarithm rules that are helpful for solving inequalities include:
Specific logarithm rules that are helpful for solving inequalities include:
- Understanding that \(\ln(x)\) is only defined for \(x > 0\).
- Knowing that \(\ln(x)\) increases as \(x\) increases, which means if \(x > y\) then \(\ln(x) > \ln(y)\), for positive \(x\) and \(y\).
- Recognizing that for \(x > 0\), \(\ln(x) \< x\), with equality if and only if \(x = 1\).
Other exercises in this chapter
Problem 2
For \(a>0\), show that $$ \lim _{n \rightarrow \infty} n\left[a^{1 / n}-1\right]=\ln a $$
View solution Problem 2
Find the unique solution of the differential equation $$ \left\\{\begin{array}{l} F^{\prime}(x)=x / \sqrt{1-x^{4}}, \quad-1
View solution Problem 3
Derive formulas for \(\cos (a-b)\) and \(\sin (a-b)\) in terms of \(\sin a, \sin b, \cos a,\) and \(\cos b\)
View solution Problem 3
Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) are periodic functions of period \(T .\) Under what condition
View solution