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Chapter 1: Problem 2

$$ \ln \sqrt{e} $$

Short Answer

Expert verified
\( \ln \sqrt{e} = \frac{1}{2} \).

Step by step solution

01

Express the Square Root as an Exponent

The first step is to express the square root using exponents. Recall that the square root of a number is the same as raising the number to the power of \( \frac{1}{2} \). Thus, we can rewrite \( \sqrt{e} \) as \( e^{\frac{1}{2}} \).
02

Apply Logarithm Power Rule

The logarithm power rule states that \( \ln(a^b) = b \cdot \ln(a) \). We can apply this rule to \( \ln(e^{\frac{1}{2}}) \) by moving the exponent \( \frac{1}{2} \) to the front of the logarithm as a multiplier. Thus, \( \ln(e^{\frac{1}{2}}) \) becomes \( \frac{1}{2} \cdot \ln(e) \).
03

Evaluate \( \ln(e) \)

The natural logarithm of \( e \) is \( 1 \), because \( \ln(e) \) represents the power to which \( e \) must be raised to obtain \( e \), which is 1. Therefore, \( \ln(e) = 1 \).
04

Calculate the Final Expression

Substitute \( \ln(e) = 1 \) back into the expression from Step 2. We have \( \frac{1}{2} \cdot 1 \), which simplifies to \( \frac{1}{2} \).

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Key Concepts

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

Logarithm Power Rule
The logarithm power rule is a fundamental tool in simplifying expressions that involve logarithms. This rule states that the logarithm of a power can be rewritten by bringing the exponent in front of the logarithm as a multiplier. Specifically, it can be expressed as \( \ln(a^b) = b \cdot \ln(a) \). By using this property, we can simplify expressions like \( \ln(e^{1/2}) \) effectively.
Here's how it works in our context:
  • Identify the exponent, which is \( \frac{1}{2} \) in \( e^{1/2} \).
  • Bring \( \frac{1}{2} \) in front of \( \ln(e^{1/2}) \) to simplify to \( \frac{1}{2} \cdot \ln(e) \).
Using this rule not only makes calculations easier but also helps in understanding the underlying relationships within logarithmic expressions. It's crucial for breaking down complex problems into more manageable parts.
Exponentiation
Exponentiation is the process of raising a number, known as the base, to a power or an exponent. In this problem, we deal with \( e^{1/2} \), which represents the square root of \( e \). Remember the relationship between roots and exponents:
  • The square root of a number is the same as raising the number to the power of \( \frac{1}{2} \).
  • Thus, \( \sqrt{e} \) becomes \( e^{\frac{1}{2}} \).
This transformation allows us to apply other mathematical properties, such as the logarithm power rule, to simplify calculations. Understanding exponentiation is key to manipulating and simplifying expressions involving powers in both algebra and calculus.
Simplifying Expressions
Simplifying expressions is a process that involves reducing them to their simplest form. In this scenario, we started with \( \ln(\sqrt{e}) \) and needed to simplify to achieve a straightforward expression.
This process included several steps:
  • First, transform the square root into an exponent (\( e^{\frac{1}{2}} \)).
  • Then, apply the logarithm power rule to bring the exponent in front, resulting in \( \frac{1}{2} \cdot \ln(e) \).
  • Lastly, recognize that \( \ln(e) \) equals 1, as \( e^1 = e \), which simplifies our expression to \( \frac{1}{2} \cdot 1 = \frac{1}{2} \).
Through these steps, we convert more complex expressions into a form that is easier to understand and use. Simplifying expressions is crucial for solving mathematical problems efficiently and accurately.

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Most popular questions from this chapter

Let \((2,1),(-3,-2)\), and \((a, b)\) form a triangle. Show that the collection of points \((a, b)\) for which the triangle is isosceles, and for which \((a, b)\) is the vertex common to the two sides of equal length, is a line (with one point deleted). Find an equation of that line.

Approximate all zeros of the function to the nearest hundredth. Let \(f(x)=a x^{2}+b x+c\) with \(a \neq 0\). Suppose \(f\) has two zeros, \(z_{1}\) and \(z_{2}\). Express \(z_{1}+z_{2}\) in terms of \(a, b\), and \(c\).

Show that the midpoints of the sides of any rectangle are the vertices of a rhombus (a quadrilateral with all sides of equal length). (Hint: Let the vertices of the rectangle be \((0,0),(a, 0),(0, b)\), and \((a, b) .)\)

A tank has the form of a right circular cylinder with hemispherical ends. Its volume \(V\) is 100 cubic meters. a. Find the length \(L\) of the cylinder in terms of the radius of the hemispheres. b. Find the length \(L\) (to the nearest centimeter) if the radius of the hemispheres is 2 meters. c. How much longer (to the nearest centimeter) would the cylindrical portion of the tank need to be if the radius of the hemispheres were 1 meter instead of 2 meters?

Suppose that \(f\) is a function and that \(g(x)=f(x)+d\) for all \(x\) in the domain of \(f\), where \(d\) is a constant. What is the relationship between the graphs of \(f\) and \(g\) ?
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