Find the exact value of the logarithmic expression $\mathrm{ln}\left(\frac{1}{\sqrt{e}}\right)$ without using a calculator.If this is not possible, state the reason.

Globokim8
2021-11-09
Answered

Find the exact value of the logarithmic expression $\mathrm{ln}\left(\frac{1}{\sqrt{e}}\right)$ without using a calculator.If this is not possible, state the reason.

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Faiza Fuller

Answered 2021-11-10
Author has **108** answers

Step 1

Given:

Let$y=\mathrm{ln}\left(\frac{1}{\sqrt{e}}\right)$

To Find: To evaluate the given expression.

Step 2

Solution:

$y=\mathrm{ln}\left(\frac{1}{\sqrt{e}}\right)$

$\text{We know that}\text{}\mathrm{ln}\left(\frac{a}{b}\right)=\mathrm{ln}a-\mathrm{ln}b\text{}\text{}\text{}\text{}\text{}\text{}\text{(Properties of logarithms)}$

$\Rightarrow y=\mathrm{ln}1-\mathrm{ln}\left(\sqrt{e}\right)$

$\text{Since}\text{}\mathrm{ln}1=0$

$\Rightarrow y=0-\mathrm{ln}\left(\sqrt{e}\right)$

$y=-{\mathrm{ln}\left(e\right)}^{\frac{1}{2}}$

$\text{We know that}\text{}{\mathrm{ln}x}^{n}\text{}\text{}\text{}\text{}\text{}\text{}\text{(Properties of logarithms)}$

$\Rightarrow y=-\frac{1}{2}\mathrm{ln}e$

$\text{Since}\text{}\mathrm{ln}e=1$

$\Rightarrow y=-\frac{1}{2}\times 1$

$y=\frac{-1}{2}$

$\text{Hence}\text{}\mathrm{ln}\left(\frac{1}{\sqrt{e}}\right)=\frac{-1}{2}$

Given:

Let

To Find: To evaluate the given expression.

Step 2

Solution:

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