To solve [tex]\(\log_3 \frac{1}{\sqrt{27}}\)[/tex], we can follow several steps using the properties of logarithms and exponents.
1. Simplifying the Expression Inside the Logarithm:
First, note that [tex]\(27\)[/tex] can be written as a power of 3:
[tex]\[
27 = 3^3
\][/tex]
Next, let's rewrite [tex]\(\frac{1}{\sqrt{27}}\)[/tex] in terms of powers of 3. The square root of 27 is:
[tex]\[
\sqrt{27} = \sqrt{3^3} = 3^{3/2}
\][/tex]
Thus, [tex]\(\frac{1}{\sqrt{27}}\)[/tex] can be expressed as:
[tex]\[
\frac{1}{\sqrt{27}} = \frac{1}{3^{3/2}} = 3^{-3/2}
\][/tex]
2. Applying the Logarithm Properties:
Now we need to find [tex]\(\log_3 (3^{-3/2})\)[/tex]. To do this, we use the property of logarithms that states:
[tex]\[
\log_b (a^c) = c \cdot \log_b (a)
\][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( c = -3/2 \)[/tex], and the base [tex]\( b = 3 \)[/tex]. Applying this property:
[tex]\[
\log_3 (3^{-3/2}) = -\frac{3}{2} \cdot \log_3 (3)
\][/tex]
3. Evaluating Basic Logarithm:
We know that [tex]\(\log_3 (3) = 1\)[/tex] because the logarithm of a number with its own base is always 1.
4. Final Calculation:
Substituting [tex]\(\log_3 (3) = 1\)[/tex] into our equation:
[tex]\[
\log_3 (3^{-3/2}) = -\frac{3}{2} \cdot 1 = -\frac{3}{2}
\][/tex]
Thus, the value of [tex]\(\log_3 \frac{1}{\sqrt{27}}\)[/tex] is [tex]\(-1.5\)[/tex].
