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Chapter 4: Problem 71

Evaluate each expression without using a calculator. $$\ln \frac{1}{e^{6}}$$

Short Answer

Expert verified
-6

Step by step solution

01

Exponentiate the denominator

This problem includes a logarithm with a fraction where the denominator is an exponential function, in particular an exponential function where the base is e since \(e^{6}\). The first step involves simplifying this. Remember that \(e\) raised to any power inside a natural logarithm can be simplified since the natural logarithm (ln) is a logarithm to the base \(e\). This means that \(ln(e^{6})\) simplifies to 6.
02

Apply Logarithm Properties

The natural logarithm of a reciprocal is equal to the negative of the natural logarithm of the number. That is, \(\ln(1/n) = -\ln(n)\). This rule can be applied to the original problem, yielding \(-\ln(e^{6})\). Now, use the simplification from Step 1 that \(\ln(e^{6}) = 6\) to find that the given expression is equal to \(-6\).

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Most popular questions from this chapter

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \sqrt[3]{\frac{x}{y}} $$

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; 1-\frac{1}{2}+\frac{1}{3} ; 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} ; \ldots .\) Describe what you observe.

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human car. The formula $$ D=10\left(\log I-\log I_{0}\right) $$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter". and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 8 \ln (x+9)-4 \ln x $$
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