Simplify ln((1)/(sqrte)) Answer:

Identify Given Expression: Identify the given expression and the properties of logarithms that can be applied. The given expression is ln((1)/(√e)). We can use the property that ln(1/x) = -ln(x) and that ln(x^(1/2)) = (1/2)ln(x).Apply Logarithm Property: Apply the logarithm property to the expression. Using the property ln(1/x) = -ln(x), we can rewrite ln((1)/(√e)) as -ln(√e).Simplify Inside Logarithm: Simplify the expression inside the logarithm. Since √e is the same as e^(1/2), we can rewrite -ln(√e) as -ln(e^(1/2)).Apply Power Rule: Apply the logarithm power rule. Using the power rule ln(x^(a)) = a*ln(x), we can simplify -ln(e^(1/2)) to -(1/2)ln(e).Simplify Logarithm: Simplify the logarithm of the base e. Since ln(e) is equal to 1, we can simplify -(1/2)ln(e) to -(1/2)*1.Calculate Final Value: Calculate the final value. Multiplying -(1/2) by 1 gives us -1/2.

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Q. Simplify

\ln \left(\frac{1}{\sqrt{e}}\right)

\newline

Answer:

Identify Given Expression: Identify the given expression and the properties of logarithms that can be applied. $\newline$ The given expression is $\ln\left(\frac{1}{\sqrt{e}}\right)$ . We can use the property that $\ln\left(\frac{1}{x}\right) = -\ln(x)$ and that $\ln\left(x^{\frac{1}{2}}\right) = \frac{1}{2}\ln(x)$ .
Apply Logarithm Property: Apply the logarithm property to the expression. $\newline$ Using the property $\ln(\frac{1}{x}) = -\ln(x)$ , we can rewrite $\ln(\frac{1}{\sqrt{e}})$ as $-\ln(\sqrt{e})$ .
Simplify Inside Logarithm: Simplify the expression inside the logarithm. $\newline$ Since $\sqrt{e}$ is the same as $e^{(1/2)}$ , we can rewrite $-\ln(\sqrt{e})$ as $-\ln(e^{(1/2)})$ .
Apply Power Rule: Apply the logarithm power rule. $\newline$ Using the power rule $\ln(x^{a}) = a\cdot\ln(x)$ , we can simplify $-\ln(e^{\frac{1}{2}})$ to $-\left(\frac{1}{2}\right)\ln(e)$ .
Simplify Logarithm: Simplify the logarithm of the base $e$ . $\newline$ Since $\ln(e)$ is equal to $1$ , we can simplify $-(\frac{1}{2})\ln(e)$ to $-(\frac{1}{2})\cdot 1$ .
Calculate Final Value: Calculate the final value. $\newline$ Multiplying $-\frac{1}{2}$ by $1$ gives us $-\frac{1}{2}$ .