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Chapter 5: Problem 48

Simplify the expression. $$\text { ln } \sqrt{e}$$

Short Answer

Expert verified
The simplified expression is \( \frac{1}{2} \).

Step by step solution

01

Express the square root as an exponent

The expression contains a square root, which can be rewritten using exponents. The square root of a number is the same as raising it to the power of 1/2. Thus, \( \sqrt{e} \) can be rewritten as \( e^{1/2} \).
02

Apply the logarithm power rule

Next, recognize that the natural logarithm of an expression raised to a power can be simplified using the logarithm power rule: \( \ln(a^b) = b \ln(a) \). Apply this rule to \( \ln(e^{1/2}) \). According to the rule, this is equal to \( \frac{1}{2} \ln(e) \).
03

Simplify using the natural logarithm of e

The natural logarithm function \( \ln(e) \) is simply 1 since \( e \) is the base of natural logarithms. Therefore, \( \frac{1}{2} \ln(e) \) simplifies to \( \frac{1}{2} \times 1 = \frac{1}{2} \).

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Key Concepts

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

Exponents
Exponents are a way to represent repeated multiplication of the same number. For instance, when we say \( e^{1/2} \), it is the same as saying we want the square root of \( e \). Let's break it down:
  • When any number 'a' is raised to a power 'b', denoted as \( a^b \), it means 'a' is multiplied by itself 'b' times.
  • The expression \( \sqrt{e} \) is read as "the square root of e," which can be rewritten as \( e^{1/2} \).
  • This transformation is handy because it helps us use exponent rules to simplify complex expressions.
Using exponents often makes equations easier to manage, especially when other mathematical tools like logarithms come into play.
Remember, any base raised to the power of 1/2 translates to its square root. This is a basic rule of exponents and widely utilized in mathematical simplifications.
Logarithmic functions
Logarithmic functions are the inverse operations of exponentiation. Logarithms tell us what power we must raise a given base to produce a certain number. The core properties of logarithms make them especially useful for simplifying expressions involving exponents.
  • The formula \( \log_a(b) \) represents the power to which the base 'a' must be raised to get the number 'b'.
  • The inverse nature helps if you initially have an exponential equation and need to bring it to a simpler form via logarithms.
  • One key rule is the logarithm power rule: \( \ln(a^b) = b \ln(a) \). This means if you have an expression with a power, you can bring the power out front as a multiplication factor.
This helps in simplifying expressions like \( \ln(e^{1/2}) \) where the power is easily moved in front of the logarithm, making calculations straightforward. Understanding these basic logarithmic properties is crucial when dealing with expressions that involve both exponents and logarithms.
Natural logarithms
Natural logarithms, commonly denoted as \( \ln \), feature a specific base, the number \( e \), approximately equal to 2.71828. This base is natural due to its frequent appearance in mathematics, particularly in calculus and differential equations.
  • The value \( \ln(e) \) is equal to 1. This is because \( e \) raised to the power of 1 is \( e \).
  • This property allows for significant simplifications in expressions involving \( e \) and natural logarithms, as seen in the example \( \ln(e^{1/2}) \).
  • Naturally, the function \( \ln(x) \) gives us the power needed to raise \( e \) in order to achieve the number \( x \).
These simple but powerful properties allow natural logarithms to play a pivotal role in various mathematical calculations. Recognizing that \( \ln(e) = 1 \) allows quick simplification and is essential for mastering mathematical expressions dealing with natural bases.

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Most popular questions from this chapter

The number of females working in automotive repair is increasing. The table shows the number of female ASE-certified technicians for selected years. $$\begin{array}{ccccc}\text { Year } & 1988 & 1989 & 1990 & 1991 \\ \hline \text { Total } & 556 & 614 & 654 & 737\end{array}$$ $$\begin{array}{|ccccc}\hline \text { Year } & 1992 & 1993 & 1994 & 1995 \\\\\hline \text { Total } & 849 & 1086 & 1329 & 1592\end{array}$$ (a) What type of function might model these data? (b) Use least-squares regression to find an exponential function given by \(f(x)=a b^{x}\) that models the data. Let \(x=0\) correspond to 1988 (c) Use \(f\) to estimate the number of certified female technicians in \(2005 .\) Round the result to the nearest hundred.

Graph \(f\) and state its domain. $$f(x)=\log (x+1)$$

Cooling an Object \(A\) pot of boiling water with a temperature of \(100^{\circ} \mathrm{C}\) is set in a room with a temperature of \(20^{\circ} \mathrm{C}\). The temperature \(T\) of the water after \(x\) hours is given by \(T(x)=20+80 e^{-x}\) (a) Estimate the temperature of the water after 1 hour. (b) How long did it take the water to cool to \(60^{\circ} \mathrm{C} ?\)

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$3\left(10^{x-2}\right)=72$$

Graph \(y=f(x)\). Is \(f\) increasing or decreasing on its domain? $$f(x)=\log _{1 / 3} x$$
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