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Chapter 4: Problem 72

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

Short Answer

Expert verified
The result of the expression \(\ln \frac{1}{e^{7}}\) is -7.

Step by step solution

01

Identification

Identify the number \(\frac{1}{e^{7}}\) as \(e^{-7}\). This because, taking any number to the negative power is equivalent to taking the reciprocal of that number to that power.
02

Apply log law

Applying the logarithm law \(\ln(a^{-n}) = -n\ln(a)\), we substitute \(-7\) for \(n\) and \(e\) for \(a\). Hence, \(\ln \frac{1}{e^{7}} = \ln (e^{-7}) = -7\ln(e)\).
03

Simplify

\(\ln(e)\) is the logarithm base \(e\) of \(e\), which is equal to 1, hence, \(-7\ln(e) = -7*1 = -7\).

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Most popular questions from this chapter

If \(f(x)=\log _{\phi} x,\) show that \(\frac{f(x+h)-f(x)}{h}=\log _{b}\left(1+\frac{h}{x}\right)^{1 / h}, h \neq 0\)

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