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Chapter 12: Problem 13

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\ln \left(\frac{e^{2}}{5}\right)$$

Short Answer

Expert verified
The simplified form of the log expression is \(2 - \ln(5)\)

Step by step solution

01

Express the fraction inside the log as a difference

Recall that the logarithm of a quotient can be expressed as a difference of logarithms. Here, \(\ln \left(\frac{e^{2}}{5}\right)\) can be written as \(\ln(e^{2}) - \ln(5)\)
02

Simplify the Logarithm of \(e^{2}\)

By the property of the natural logarithm, \(\ln(e^{x}) = x\). Therefore, \(\ln(e^{2}) = 2\). So our expression becomes \(2 - \ln(5)\)
03

Final Expression

The final simplified form of the given log expression is \(2 - \ln(5)\). With no further simplification possible, this is the answer.

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Most popular questions from this chapter

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\)

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 17,425 & 4.25 \% & \$ 25,000 & \\ \hline \end{array}$$

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 2350 & & \text { Triple the amount invested } & 7 \\ \hline \end{array}$$

Solve: $$\frac{3}{x+1}-\frac{5}{x}=\frac{19}{x^{2}+x}$$

Solve each equation. $$3^{x^{2}-12}=9^{2 x}$$
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