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Chapter 10: Problem 6

Without using a calculator, give the value of \(\ln e^{\sqrt{3}}\)

Short Answer

Expert verified
\(\sqrt{3}\)

Step by step solution

01

Understand the properties of natural logarithms and exponents

The natural logarithm function, \( ln(x) \), and the exponential function, \( e^x \), are inverses of each other. This means that \( ln(e^x) = x \). Knowing this property will help solve the problem.
02

Apply the property to the given expression

Given \( ln(e^{ \sqrt{3}}) \), apply the property of logarithms: \( ln(e^x) = x \). Thus, \( ln(e^{ \sqrt{3}}) = \sqrt{3} \).

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Key Concepts

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

properties of natural logarithms
The natural logarithm, denoted as \(\ln(x)\), is a special logarithm with base \(\mathit{e}\), where \(\mathit{e}\) is an irrational and transcendental number approximately equal to 2.71828. One key property to remember is that the natural logarithm and the exponential function are inverses of each other. This can be written as \[ \ln(e^x) = x \], where \(\mathit{e}\) is the base of the natural logarithms.
exponential functions
An exponential function takes the form \(e^x\), where \(e\) is Euler's number. This function grows very quickly because each step involves multiplying by \(e\). The natural logarithm undoes this growth, pulling the exponent back down. So when you see \(ln(e^{number})\), you know it simplifies to just 'number'. For example, \(\ln(e^{\sqrt{3}}) = \sqrt{3}\).
inverse functions
Inverse functions essentially 'cancel' each other out. The exponential function \(e^x\) and the natural logarithm \(ln(x)\) are inverses. This means \(ln(e^x) = x\) and \(e^{ln(x)} = x\). In our problem, \(ln(e^{\sqrt{3}})\), using their properties makes it a straightforward simplification to just \(\sqrt{3}\).

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Most popular questions from this chapter

A major scientific periodical published an article in 1990 dealing with the problem of global warming. The article was accompanied by a graph that illustrated two possible scenarios. (a) The warming might be modeled by an exponential function of the form $$y=\left(1.046 \times 10^{-38}\right)\left(1.0444^{x}\right)$$ (b) The warming might be modeled by a linear function of the form $$y=0.009 x-17.67$$ In both cases, \(x\) represents the year, and y represents the increase in degrees Celsius due to the warming. Use these functions to approximate the increase in temperature for each of the following years. $$ 2000 $$

Consumers can now enjoy movies at home in elaborate home-theater systems. Find the average decibel level \(D=10 \log \left(\frac{I}{I_{0}}\right)\) for each movie with the given intensity \(I\) (a) Avatar; \(5.012 \times 10^{10} \mathrm{I}_{0}\) (b) Iron Man \(2 ; \quad 10^{10} I_{0}\) (c) Clash of the Titans; \(6,310,000,000 \mathrm{I}_{0}\)

A major scientific periodical published an article in 1990 dealing with the problem of global warming. The article was accompanied by a graph that illustrated two possible scenarios. (a) The warming might be modeled by an exponential function of the form $$y=\left(1.046 \times 10^{-38}\right)\left(1.0444^{x}\right)$$ (b) The warming might be modeled by a linear function of the form $$y=0.009 x-17.67$$ In both cases, \(x\) represents the year, and y represents the increase in degrees Celsius due to the warming. Use these functions to approximate the increase in temperature for each of the following years. $$ 2010 $$

When a student asked his teacher to explain how to evaluate $$ \log _{9} 3 $$ without showing any work, his teacher told him, "Think radically." Explain what the teacher meant by this hint.

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$ 3 \log _{a} 5-\frac{1}{2} \log _{a} 9 $$
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