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Chapter 4: Problem 25

Rewrite each expression as a sum or difference of logarithms. $$\ln \left(\frac{x-1}{x}\right)$$

Short Answer

Expert verified
\[ \text{ln}(x-1) - \text{ln}(x) \]

Step by step solution

01

Identify the Logarithmic Property

Use the property of logarithms that states \(\frac{\text{ln}(\frac{a}{b}) = \text{ln}(a) - \text{ln}(b)}\). This helps to separate the logarithm of a quotient into a difference of two logarithms.
02

Rewrite the Expression

Apply the property identified in the previous step to rewrite the given expression: \[ \text{ln} \bigg(\frac{x-1}{x}\bigg) = \text{ln}(x-1) - \text{ln}(x) \]

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Key Concepts

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

logarithms
Logarithms are the inverse functions of exponentiation. They help solve equations where the exponent is unknown. For example, the expression \(\text{log}_b(a)\) answers the question: 'To what power must the base b be raised, to get a?' As an illustration, \(\text{log}_2(8) = 3\) since \({2^3} = 8\). It can be helpful to know the several types of logarithms are commonly used: the common logarithm (base 10), the binary logarithm (base 2), and the natural logarithm (base e). Each type has its own uses in various fields of study.
quotient rule
The quotient rule is a special property of logarithms that simplifies the logarithm of a quotient into the difference of two separate logarithms. If you have an expression like \(\text{ln}(\frac{a}{b})\), you can rewrite it using the quotient rule: \(\text{ln}(a) - \text{ln}(b)\). This can make mathematical operations easier to handle, especially when solving equations. The rule is particularly useful in breaking down complex logarithmic expressions into simpler components.
This rule is based on the logarithmic identity:
  • \(\text{ln}(\frac{a}{b}) = \text{ln}(a) - \text{ln}(b)\)
This key identity helps in tasks like solving and simplifying logarithmic expressions.
natural logarithm
The natural logarithm, commonly denoted as \( \text{ln} \) or \( \text{log}_e \), is the logarithm to the base e, where e is approximately equal to 2.71828. The number e is a constant that appears frequently in calculus and complex analysis. Natural logarithms are particularly useful in scenarios related to exponential growth and decay, such as population studies and radioactive decay. An example of how the natural logarithm works:
  • \(\text{ln}(e) = 1\) since \(e^1=e\)

The natural logarithm has properties that make it convenient for various mathematical operations, such as integration and differentiation in calculus.
logarithmic expressions
Logarithmic expressions involve logarithms and can be simplified using various properties like the product rule, quotient rule, and power rule. For instance, the given exercise asks you to rewrite \( \text{ln} \bigg(\frac{x-1}{x}\bigg) \) as a sum or difference of logarithms. By using the quotient rule, you can transform it into:
  • \(\text{ln}(x-1) - \text{ln}(x)\)

This process helps break complex expressions into simpler parts, making them easier to manage and solve. Logarithmic expressions are key elements in higher-level math and science courses.

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Most popular questions from this chapter

Solve each equation. Find the exact solutions. $$\log (x)=\log \left(6-x^{2}\right)$$

The graph of \(y=2^{x}\) is reflected in the \(x\) -axis, translated 5 units to the right, and then 9 units upward. Find the equation of the curve in its final position.

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