To solve the equation [tex]\(\log_9 \sqrt{27} = x\)[/tex], let's break it down step-by-step.
1. Simplify the expression [tex]\(\sqrt{27}\)[/tex]:
The square root of 27 can be written as [tex]\(27^{1/2}\)[/tex].
2. Rewrite 27 in terms of base 3:
Since [tex]\(27 = 3^3\)[/tex], we can write:
[tex]\[
27^{1/2} = (3^3)^{1/2} = 3^{3/2}
\][/tex]
3. Substitute [tex]\(\sqrt{27}\)[/tex] with [tex]\(3^{3/2}\)[/tex] in the logarithmic expression:
We now have:
[tex]\[
\log_9 (3^{3/2}) = x
\][/tex]
4. Rewrite the log expression using the change of base property:
The change of base formula for logarithms states:
[tex]\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\][/tex]
Let's apply the natural logarithm (base [tex]\(e\)[/tex]) here:
[tex]\[
\log_9 (3^{3/2}) = \frac{\log(3^{3/2})}{\log(9)}
\][/tex]
5. Simplify the numerator and the denominator:
Using the property of logarithms [tex]\(\log(a^b) = b \log(a)\)[/tex], we get:
[tex]\[
\log(3^{3/2}) = \frac{3}{2} \log(3)
\][/tex]
And since [tex]\(9 = 3^2\)[/tex]:
[tex]\[
\log(9) = \log(3^2) = 2 \log(3)
\][/tex]
So, we substitute back into our expression:
[tex]\[
\log_9 (3^{3/2}) = \frac{\frac{3}{2} \log(3)}{2 \log(3)}
\][/tex]
6. Simplify the fraction:
Cancel out [tex]\(\log(3)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{\frac{3}{2} \log(3)}{2 \log(3)} = \frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}
\][/tex]
Therefore, the solution to the equation [tex]\(\log_9 \sqrt{27} = x\)[/tex] is:
[tex]\[
x = \frac{3}{4}
\][/tex]
So the correct answer is:
c. [tex]\(3 / 4\)[/tex]
