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Chapter 4: Problem 81

Use the properties of natural logarithms to rewrite the expression. $$\ln e^{2}$$

Short Answer

Expert verified
The simplified form of the given expression is 2.

Step by step solution

01

Apply logarithmic property

We know that if we have \(\ln a^b\), it can be written as \(b \ln a\). Using this property, we can simplify the given expression as \(2 \ln e\).
02

Use logarithm identity

The natural logarithm of \(e\) is 1. So, \(2 \ln e\) becomes \(2 \times 1 = 2\). After this step, the entire problem is solved.

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Key Concepts

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

Logarithmic Properties
Logarithms are an integral part of mathematics, particularly when dealing with exponential equations. Understanding logarithmic properties is fundamental in simplifying complex expressions into more manageable forms.

The most crucial logarithmic properties to remember include:
  • The product rule: \( \ln(ab) = \ln(a) + \ln(b) \), which allows us to separate logarithms of multiplications into sums.
  • The quotient rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \), which enables us to write the logarithm of a division as the difference of logarithms.
  • The power rule: \( \ln(a^b) = b\ln(a) \), which is essential for handling powers within logarithms by bringing the exponent out as a coefficient.
Applying these properties can break down complex expressions and provide a pathway to solving logarithmic equations. For example, in the textbook exercise, by identifying and applying the power rule \( \ln(e^2) = 2\ln(e) \) the expression becomes simpler to resolve.
Natural Logarithm Identities
The natural logarithm, represented as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. A deep understanding of the natural logarithm identities is key to mastering the art of simplifying logarithmic expressions.

Some of the most pivotal identities include:
  • \(\ln(1) = 0\): The log of 1 to any base is always 0 because any number raised to the power of 0 equals 1.
  • \(\ln(e) = 1\): The log of \(e\) to the base \(e\) is 1, since the definition of the logarithm is to find the exponent that the base must be raised to produce a given number.
  • \(\ln(e^x) = x\): The natural log of \(e \text{ raised to any exponent } x\) simply returns \(x\).
In the exercise provided, the identity \(\ln(e) = 1\) is used in the second step to simplify \(2\ln(e)\) to \(2\times1\) hence obtaining the answer 2.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions involves using logarithmic properties and identities to condense or rewrite expressions for ease of understanding and computation.

To effectively simplify a logarithmic expression:
  • Identify the log properties that can be applied.
  • Rewrite the equation using these properties.
  • Simplify further by substituting known logarithm values and performing basic arithmetic operations.
In practice, the exercise solution takes the given \(\ln(e^2)\) and simplifies it using the power rule, resulting in \(2\ln(e)\). Recognizing that \(\ln(e)\) is equal to 1, the expression simplifies to 2. This methodical approach can be applied to a multitude of logarithmic expressions for efficient problem-solving.

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Most popular questions from this chapter

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0.5, 1.0), (2, 12.5), (4, 33.2), (6, 65.7), (8, 98.5),(10, 150.0)$$

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(2,450),(4,385),(6,345),(8,332),(10,312)$$

Sketch the graph of the function. $$f(x)=|x-2|-8$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$4 \log _{10}(x-6)=11$$

Find the time required for a \(\$ 1000\) investment to (a) double at interest rate \(r,\) compounded continuously, and (b) triple at interest rate \(r\), compounded continuously. Round your results to two decimal places. $$r=7 \%$$
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