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Chapter 5: Problem 19

Use the properties of logarithms to expand the logarithmic expression. \(\ln \frac{2}{3}\)

Short Answer

Expert verified
The expanded form of the given logarithmic expression is \(\ln(2) - \ln(3)\)

Step by step solution

01

Identify The Logarithmic Property

The expression given is written in a form \(\ln(f/g)\). A logarithmic rule is applied here which states that \(\log_b(f / g) = \log_b f - \log_b g\). In this case, the base (b) is e because the natural logarithm (ln) is being used.
02

Apply The Logarithmic Property

Apply the logarithmic property \(\ln(f/g) = \ln(f) - \ln(g)\) to the expression \(\ln(2/3)\). Thus, the expression \(\ln(2/3)\) can be rewritten as \(\ln(2) - \ln(3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Expansion
Logarithmic expansion is a technique used in mathematics to break down complex logarithmic expressions into simpler components. By expanding these expressions, students can understand and solve problems more effectively. For example, if you have an expression like \( \ln \frac{a}{b} \), you can expand it using the logarithmic quotient rule. This rule states that \( \ln \frac{a}{b} = \ln(a) - \ln(b) \).
  • This helps to separate the numerator and denominator, making calculations more straightforward.
  • It is especially useful in calculus and algebra where simplification of expressions is necessary.
  • This can also aid in better visualization and understanding of the problem.
By expanding logarithms, we can use simple arithmetic operations to assess and understand the relationships between numbers in expressions.
Natural Logarithm
The natural logarithm, denoted by \( \ln \), is a type of logarithm that has the base \( e \), where \( e \approx 2.718 \). It's a fundamental mathematical constant that arises in various branches of mathematics, such as calculus and complex analysis. When we see \( \ln \), it is shorthand for "logarithm base \( e \)".
  • Natural logarithms are prevalent in natural and applied sciences.
  • They often describe exponential growth and decay, such as population or radioactive decay.
  • The natural logarithm has certain properties that simplify many types of calculations.
In the context of the exercise, understanding that \( \ln \) is essentially a logarithm with base \( e \), allows us to apply various logarithmic properties to solve expressions more easily.
Logarithmic Identity
Logarithmic identities are formulas and properties that apply to logarithms, helping to simplify and manipulate logarithmic expressions. One key identity used in the example above is the quotient rule.

Key Logarithmic Identities

  • Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
  • Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \)
  • Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \)
These identities are essential tools in mathematics, allowing for greater flexibility when working with logarithmic functions. They simplify complex calculations, which would otherwise be cumbersome if left in their original form.
To tackle real-world problems, like those involving exponential relationships, mastering these identities can streamline your problem-solving process.

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Most popular questions from this chapter

Evaluate the integral. $$ \int_{0}^{\ln 2} \tanh x d x $$

Find the integral. $$ \int \frac{\cosh \sqrt{x}}{\sqrt{x}} d x $$

An object is projected upward from ground level with an initial velocity of 500 feet per second. In this exercise, the goal is to analyze the motion of the object during its upward flight. (a) If air resistance is neglected, find the velocity of the object as a function of time. Use a graphing utility to graph this function. (b) Use the result in part (a) to find the position function and determine the maximum height attained by the object. (c) If the air resistance is proportional to the square of the velocity, you obtain the equation $$ \frac{d v}{d t}=-\left(32+k v^{2}\right) $$ where \(-32\) feet per second per second is the acceleration due to gravity and \(k\) is a constant. Find the velocity as a function of time by solving the equation $$ \int \frac{d v}{32+k v^{2}}=-\int d t $$ (d) Use a graphing utility to graph the velocity function \(v(t)\) in part (c) if \(k=0.001\). Use the graph to approximate the time \(t_{0}\) at which the object reaches its maximum height. (e) Use the integration capabilities of a graphing utility to approximate the integral \(\int_{0}^{t_{0}} v(t) d t\) where \(v(t)\) and \(t_{0}\) are those found in part (d). This is the approximation of the maximum height of the object. (f) Explain the difference between the results in parts (b) and (e).

Find any relative extrema of the function. Use a graphing utility to confirm your result. $$ g(x)=x \operatorname{sech} x $$

Use the equation of the tractrix \(y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^{2}-x^{2}}, \quad a>0\). Find \(d y / d x\).
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