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Math  /  Algebra

QuestionExercises 1-18, evaluate the logarithmic expression without using a calculator.
1. log⁡44\log _{4} 4
2. log⁡61\log _{6} 1
3. log⁡232\log _{2} 32
4. log⁡381\log _{3} 81
5. log⁡5253\log _{5} \sqrt[3]{25}
6. log⁡61365\log _{6} \frac{1}{\sqrt[5]{36}}
7. log⁡103\log 10^{3}
8. log⁡10,000\log 10,000
9. log⁡100,000\log 100,000
10. log⁡10−4\log 10^{-4}
11. log⁡103\log \sqrt[3]{10}
12. log⁡11000\log \frac{1}{\sqrt{1000}}
13. ln⁡e3\ln e^{3}
14. ln⁡e−4\ln e^{-4}
15. ln⁡1e\ln \frac{1}{e}
16. ln⁡1\ln 1
17. ln⁡e4\ln \sqrt[4]{e}
18. ln⁡1e7\ln \frac{1}{\sqrt{e^{7}}}

Studdy Solution
Evaluate ln⁡1e7\ln \frac{1}{\sqrt{e^{7}}}:
Express 1e7\frac{1}{\sqrt{e^{7}}} as a power: e−7/2e^{-7/2}. Then, use the power rule of natural logarithms:
ln⁡1e7=ln⁡e−7/2=−72ln⁡e\ln \frac{1}{\sqrt{e^{7}}} = \ln e^{-7/2} = -\frac{7}{2} \ln e
Since ln⁡e=1\ln e = 1, we have:
−72×1=−72-\frac{7}{2} \times 1 = -\frac{7}{2}

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