Problem 71
Question
Evaluate each expression without using a calculator. $$\ln \frac{1}{e^{6}}$$
Step-by-Step Solution
Verified Answer
-6
1Step 1: 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.
2Step 2: 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 \(-6\).
Other exercises in this chapter
Problem 71
In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$
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Explain how to solve an exponential equation. Use \(3^{x}=140\) in your explanation.
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View solution Problem 72
In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{6} 17 $$
View solution