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Chapter 5: Problem 68

Find the following using a calculator. Round to four decimal places. $$\ln 0$$

Short Answer

Expert verified
\[\ln 0\] is undefined as it approaches negative infinity.

Step by step solution

01

Understanding the Logarithm

The natural logarithm function, denoted as \(\text{ln} x\), is the inverse of the exponential function. It is only defined for positive real numbers.
02

Evaluate \(\text{ln} 0\)

Attempting to find \(\text{ln} 0\) requires evaluating the natural logarithm of zero. However, as the natural logarithm function is undefined for non-positive values, \(\text{ln} 0\) approaches negative infinity.
03

Conclusion

Since \(\text{ln} 0\) is undefined and approaches negative infinity, the solution is that \(\text{ln} 0\) does not exist in the real number system.

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Key Concepts

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

logarithm properties
Understanding logarithm properties is crucial for solving logarithmic expressions. A logarithm answers the question: to what power must the base be raised to produce a given number? In particular, the natural logarithm, denoted as \(\text{ln} x\), uses the base \(\text{e}\), where \(\text{e}\) is approximately 2.71828.

The properties of logarithms make them easier to handle, especially when dealing with complex equations. Some key properties include:
  • Product Property: \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\)
  • Quotient Property: \(\text{ln}(a/b) = \text{ln}(a) - \text{ln}(b)\)
  • Power Property: \(\text{ln}(a^b) = b \text{ln}(a)\)
Applying these properties correctly can simplify your calculations and improve your overall understanding of logarithmic functions.
exponential function
The exponential function and the natural logarithm have a unique relationship. The exponential function, expressed as \(\text{e}^x\), is the inverse of the natural logarithm. This means that \(\text{ln}(\text{e}^x) = x\) and \(\text{e}^{\text{ln}(x)} = x\) for positive values of \(\text{x}\).

This reciprocal relationship allows us to switch between exponential and logarithmic forms seamlessly. For example, solving \(\text{e}^x = y\) for \(\text{x}\) requires taking the natural logarithm of both sides, leading to \(\text{ln}(y) = x\). Conversely, solving \(\text{ln}(x) = y\) for \(\text{x}\) involves raising \(\text{e}\) to the power of \(\text{y}\), giving us \(\text{e}^y = x\).

Utilizing this relationship helps solve equations involving exponentials and logarithms.
undefined mathematical expressions
Undefined mathematical expressions can often be challenging for students. An expression becomes undefined when it does not fit within the rules of the mathematical function being used.

For example, in the case of \(\text{ln} 0,\) it's essential to remember that the natural logarithm function is only defined for positive real numbers. Trying to evaluate \(\text{ln} 0\) leads to an undefined expression because there is no power to which \(\text{e}\) can be raised to yield zero.

This also applies to other logarithmic and exponential functions. For instance, \(\text{ln}(-x)\) is undefined for negative numbers since the base of a logarithm must always be a positive real number.

Understanding these limitations can help prevent errors and ensure that you interpret results correctly when solving mathematical problems.

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Most popular questions from this chapter

Solve using any method. Given that \(a=\log _{8} 225\) and \(b=\log _{2} 15,\) express as a function of \(b\).

Solve. $$x^{3}+6 x^{2}-16 x=0$$

The following formula can be used to convert Fahrenheit temperatures \(x\) to Celsius temperatures \(T(x):\) $$ T(x)=\frac{5}{9}(x-32) $$ a) Find \(T\left(-13^{\circ}\right)\) and \(T\left(86^{\circ}\right)\) b) Find \(T^{-1}(x)\) and explain what it represents.

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log _{3} x+\log _{3}(x+1)=\log _{3} 2+\log _{3}(x+3)$$

E-Cigarette SE-Cigarette Sales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) millionales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) million in 2012 (Sources: UBS; forbes, \(\mathrm{com}\) ). The exponential function $$ S(x)=20.913(2.236)^{x} $$ where \(x\) is the number of years after \(2008,\) models the sales, in millions of dollars. Use this function to estimate the sales of e-cigarettes in 2011 and in 2015 . Round to the nearest million dollars.
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