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Express the given expression as an integer or as a fraction in simplest form. ln((1)/(sqrte)) Answer:

Express the given expression as an integer or as a fraction in simplest form.\newlineln⁡(1e) \ln \left(\frac{1}{\sqrt{e}}\right) \newlineAnswer:

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Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.\newlineln⁡(1e) \ln \left(\frac{1}{\sqrt{e}}\right) \newlineAnswer:
  1. Understand the expression: Understand the given expression.\newlineWe need to simplify the natural logarithm of the reciprocal of the square root of ee, which is ln⁡(1e)\ln\left(\frac{1}{\sqrt{e}}\right).
  2. Apply logarithm properties: Apply the properties of logarithms.\newlineThe natural logarithm of a reciprocal is the negative of the natural logarithm of the original value. Also, the natural logarithm of the square root of a number is one-half the natural logarithm of the number itself.\newlineln⁡(1e)=−ln⁡(e)=−12×ln⁡(e)\ln\left(\frac{1}{\sqrt{e}}\right) = -\ln(\sqrt{e}) = -\frac{1}{2} \times \ln(e)
  3. Simplify ln⁡(e)\ln(e): Simplify the natural logarithm of ee.\newlineSince the natural logarithm of ee is 11 (ln⁡(e)=1\ln(e) = 1), we can simplify the expression further.\newline−12×ln⁡(e)=−12×1=−12-\frac{1}{2} \times \ln(e) = -\frac{1}{2} \times 1 = -\frac{1}{2}

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