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Chapter 4: Problem 92

Evaluate or simplify each expression without using a calculator. $$\ln \frac{1}{e^{7}}$$

Short Answer

Expert verified
Simplified result for \( \ln \frac{1}{e^{7}} \) is -7.

Step by step solution

01

Understand The Meaning of Natural Logarithm

At first, let's recall that \(\ln x\), the natural logarithm, is the power to which \(\textit{e}\) would have to be raised to equal \(\textit{x}\). In other words, if: \(\textit{e}^y = x, then \ln x = y\). This property is fundamental for solving the given problem.
02

Apply the Logarithm Properties

Next, apply logarithmic property: \(\ln{\frac{1}{a}} = -\ln{a} \). So, simplify the given expression as: \(\ln \frac{1}{e^{7}} = -\ln e^{7}\). However, applying the property of logarithms that says \(\ln{x^n} = n \ln x\), simplify to: \(-\ln e^{7} = -7 \ln e\).
03

Simplify the Logarithm

Finally, \(\ln e\) equals 1 because it refers to the power to which \(\textit{e}\) raises to get \(\textit{e}\), which is obviously 1. Thus, -7 times 1 results into -7. So, \(-7 \ln e = -7\).

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\).

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

What question can be asked to help evaluate \(\log _{3} 81 ?\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$
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