To solve the expression [tex]\(\log_2 \sqrt{2} + \log_3 \sqrt{3}\)[/tex], we'll work with each logarithm separately and then sum the results. Here's the detailed step-by-step solution:
1. Evaluate [tex]\(\log_2 \sqrt{2}\)[/tex]:
We can use the properties of logarithms and exponents to simplify [tex]\(\log_2 \sqrt{2}\)[/tex]:
[tex]\[
\log_2 \sqrt{2} = \log_2 (2^{1/2})
\][/tex]
Using the property of logarithms [tex]\(\log_b (a^c) = c \log_b a\)[/tex], we get:
[tex]\[
\log_2 (2^{1/2}) = \frac{1}{2} \log_2 2
\][/tex]
Since [tex]\(\log_2 2 = 1\)[/tex], this simplifies to:
[tex]\[
\frac{1}{2} \log_2 2 = \frac{1}{2} \cdot 1 = 0.5
\][/tex]
2. Evaluate [tex]\(\log_3 \sqrt{3}\)[/tex]:
Similarly, simplify [tex]\(\log_3 \sqrt{3}\)[/tex] using the same properties:
[tex]\[
\log_3 \sqrt{3} = \log_3 (3^{1/2})
\][/tex]
Applying the property [tex]\(\log_b (a^c) = c \log_b a\)[/tex], we get:
[tex]\[
\log_3 (3^{1/2}) = \frac{1}{2} \log_3 3
\][/tex]
Since [tex]\(\log_3 3 = 1\)[/tex], this simplifies to:
[tex]\[
\frac{1}{2} \log_3 3 = \frac{1}{2} \cdot 1 = 0.5
\][/tex]
3. Sum the results:
Now, we need to add the two logarithmic values we computed:
[tex]\[
\log_2 \sqrt{2} + \log_3 \sqrt{3} = 0.5 + 0.5 = 1.0
\][/tex]
Therefore, the value of [tex]\(\log_2 \sqrt{2} + \log_3 \sqrt{3}\)[/tex] is [tex]\(1.0\)[/tex].
To summarize:
[tex]\[
\log_2 \sqrt{2} = 0.5
\][/tex]
[tex]\[
\log_3 \sqrt{3} = 0.5
\][/tex]
[tex]\[
\log_2 \sqrt{2} + \log_3 \sqrt{3} = 1.0
\][/tex]
