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$Express the given expression as an integer or as a fraction in simplest form. log_(3)((1)/(sqrt3)) Answer:$

Express the given expression as an integer or as a fraction in simplest form. $\newline$ $\log _{3}\left(\frac{1}{\sqrt{3}}\right)$ $\newline$ Answer:

Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.

\newline

\log _{3}\left(\frac{1}{\sqrt{3}}\right)

\newline

Answer:

Understand Given Expression: Understand the given expression and the logarithm properties that can be applied. $\newline$ The given expression is $\log_{3}\left(\frac{1}{\sqrt{3}}\right)$ , which can be written as $\log_{3}(1/\sqrt{3})$ . We can use the property of logarithms that states $\log_{3}(a/b) = \log_{3}(a) - \log_{3}(b)$ .
Apply Logarithm Property: Apply the logarithm property to the given expression. $\newline$ Using the property from Step $1$ , we can express $\log_3(\frac{1}{\sqrt{3}})$ as $\log_3(1) - \log_3(\sqrt{3})$ .
Evaluate Logarithm of $1$ : Evaluate $\log_{3}(1)$ . The logarithm of $1$ to any base is always $0$ , so $\log_{3}(1) = 0$ .
Express Square Root as Power: Express $\sqrt{3}$ as $3^{(1/2)}$ and apply the logarithm property. $\newline$ The square root of $3$ can be written as $3$ raised to the power of $1/2$ , so $\log_{3}(\sqrt{3})$ becomes $\log_{3}(3^{(1/2)})$ .
Use Power Rule of Logarithms: Use the power rule of logarithms. The power rule states that $\log_3(a^b) = b \times \log_3(a)$ . Applying this to $\log_3(3^{1/2})$ , we get $(1/2) \times \log_3(3)$ .
Evaluate Logarithm of $3$ : Evaluate $\log_3(3)$ . The logarithm of a number to the same base is $1$ , so $\log_3(3) = 1$ . Therefore, $(1/2) \times \log_3(3) = (1/2) \times 1 = 1/2$ .
Combine Results: Combine the results from Step $3$ and Step $6$ . $\newline$ We have $\log_3(1) - \log_3(\sqrt{3}) = 0 - \frac{1}{2} = -\frac{1}{2}$ .