Solve the following equation:

[tex]\[ \log _9 \sqrt{27} = x \][/tex]

Select one:
a. [tex]\(\frac{4}{3}\)[/tex]
b. [tex]\(\frac{2}{3}\)[/tex]
c. [tex]\(\frac{3}{4}\)[/tex]
d. [tex]\(\frac{3}{2}\)[/tex]



Answer :

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]