Let's evaluate each logarithm step by step using the properties of logarithms.
1. [tex]\(\log _3 9 + \log _3 27\)[/tex]
Using the property of logarithms that states [tex]\(\log_b(a) + \log_b(b) = \log_b(a \cdot b)\)[/tex]:
[tex]\[
\log_3(9) + \log_3(27) = \log_3(9 \cdot 27)
\][/tex]
Calculate [tex]\(9 \cdot 27\)[/tex]:
[tex]\[
9 \cdot 27 = 243
\][/tex]
So we have:
[tex]\[
\log_3(243)
\][/tex]
We know that [tex]\(243 = 3^5\)[/tex], therefore:
[tex]\[
\log_3(3^5) = 5
\][/tex]
[tex]\[
\boxed{5}
\][/tex]
2. [tex]\(\log _2 8 - \log _2 4\)[/tex]
Using the property of logarithms that states [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex]:
[tex]\[
\log_2(8) - \log_2(4) = \log_2\left(\frac{8}{4}\right)
\][/tex]
Calculate [tex]\(\frac{8}{4}\)[/tex]:
[tex]\[
\frac{8}{4} = 2
\][/tex]
So we have:
[tex]\[
\log_2(2)
\][/tex]
We know that [tex]\(2 = 2^1\)[/tex], therefore:
[tex]\[
\log_2(2^1) = 1
\][/tex]
[tex]\[
\boxed{1}
\][/tex]
3. [tex]\(\log _5 5(5)^{\frac{1}{3}}\)[/tex]
Using the property of logarithms that states [tex]\(\log_b(a \cdot b) = \log_b(a) + \log_b(b)\)[/tex]:
[tex]\[
\log_5(5 \cdot 5^{\frac{1}{3}}) = \log_5(5) + \log_5(5^{\frac{1}{3}})
\][/tex]
We know that:
[tex]\[
\log_5(5) = 1
\][/tex]
And for [tex]\(\log_5(5^{\frac{1}{3}})\)[/tex], using the property [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex]:
[tex]\[
\log_5(5^{\frac{1}{3}}) = \frac{1}{3} \cdot \log_5(5) = \frac{1}{3} \cdot 1 = \frac{1}{3}
\][/tex]
Adding these together:
[tex]\[
\log_5(5) + \log_5(5^{\frac{1}{3}}) = 1 + \frac{1}{3} = \frac{4}{3}
\][/tex]
[tex]\[
\boxed{\frac{4}{3}}
\][/tex]
Thus, the correct answers are:
1. [tex]\(\log _3 9+\log _3 27=\boxed{5}\)[/tex]
2. [tex]\(\log _2 8-\log _2 4=\boxed{1}\)[/tex]
3. [tex]\(\log _5 5(5)^{\frac{1}{3}}=\boxed{\frac{4}{3}}\)[/tex]
