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Chapter 4: Problem 74

Evaluate each expression without using a calculator. $$e^{\ln 300}$$

Short Answer

Expert verified
The value of \(e^{ln 300}\) is 300.

Step by step solution

01

Understanding the properties of logs

Before diving into solving the problem, it's crucial to understand that the exponential function and the logarithm are inverse operations to each other. Therefore, for the base \(a\) and the number \(x\), \(a^{log_a(x)} = x\).
02

Apply this property to the problem

Using the rule from step 1, recognize that the base is \(e\) and the raised element is \(ln 300\). Therefore, \(e^{ln 300}\) simplifies to 300.

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Most popular questions from this chapter

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. $$ \ln \left[\frac{x^{4} \sqrt{x^{2}+3}}{(x+3)^{5}}\right] $$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{16} 57,2 $$

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. $$ \operatorname{og} \sqrt[5]{\frac{x}{y}} $$

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. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right) $$
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