QuestionJuly 14, 2025

Evaluate the following expressions.Your answers must be exact and in simplest form. (a) lne^11=square (b) e^ln4=square (c) e^lnsqrt (2)=square (d) ln((1)/(e^4))=square

Evaluate the following expressions.Your answers must be exact and in simplest form. (a) lne^11=square (b) e^ln4=square (c) e^lnsqrt (2)=square (d) ln((1)/(e^4))=square
Evaluate the following expressions.Your answers must be exact and in simplest form. (a) lne^11=square (b) e^ln4=square (c) e^lnsqrt (2)=square (d) ln((1)/(e^4))=square

Solution
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Answer

(a) 11 ### (b) 4 ### (c) \sqrt{2} ### (d) -4 Explanation 1. Simplify ln e^{11} Use the property ln(e^x) = x. So, ln e^{11} = 11. 2. Simplify e^{ln 4} Use the property e^{ln(x)} = x. So, e^{ln 4} = 4. 3. Simplify e^{ln \sqrt{2}} Use the property e^{ln(x)} = x. So, e^{ln \sqrt{2}} = \sqrt{2}. 4. Simplify ln(\frac{1}{e^4}) Use the property ln(\frac{1}{x}) = -ln(x). So, ln(\frac{1}{e^4}) = -ln(e^4) = -4.

Explanation

1. Simplify $ln e^{11}$<br /> Use the property $ln(e^x) = x$. So, $ln e^{11} = 11$.<br /><br />2. Simplify $e^{ln 4}$<br /> Use the property $e^{ln(x)} = x$. So, $e^{ln 4} = 4$.<br /><br />3. Simplify $e^{ln \sqrt{2}}$<br /> Use the property $e^{ln(x)} = x$. So, $e^{ln \sqrt{2}} = \sqrt{2}$.<br /><br />4. Simplify $ln(\frac{1}{e^4})$<br /> Use the property $ln(\frac{1}{x}) = -ln(x)$. So, $ln(\frac{1}{e^4}) = -ln(e^4) = -4$.
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