To simplify the expression [tex]\(\log_a(\sqrt{2}) + \log_a(8) + \log_a\left(\frac{1}{2}\right)\)[/tex] and express it in the form [tex]\(p \log_a(2)\)[/tex], we can use logarithm properties. Let's break it down step-by-step:
1. Simplify [tex]\(\log_a(\sqrt{2})\)[/tex]:
- Recall that [tex]\(\sqrt{2} = 2^{1/2}\)[/tex].
- By the power rule of logarithms, [tex]\(\log_a(2^{1/2}) = \frac{1}{2} \log_a(2)\)[/tex].
- So, [tex]\(\log_a(\sqrt{2}) = \frac{1}{2} \log_a(2)\)[/tex].
2. Simplify [tex]\(\log_a(8)\)[/tex]:
- Recall that [tex]\(8 = 2^3\)[/tex].
- By the power rule of logarithms, [tex]\(\log_a(2^3) = 3 \log_a(2)\)[/tex].
- So, [tex]\(\log_a(8) = 3 \log_a(2)\)[/tex].
3. Simplify [tex]\(\log_a\left(\frac{1}{2}\right)\)[/tex]:
- Recall that [tex]\(\frac{1}{2} = 2^{-1}\)[/tex].
- By the power rule of logarithms, [tex]\(\log_a(2^{-1}) = -\log_a(2)\)[/tex].
- So, [tex]\(\log_a\left(\frac{1}{2}\right) = -\log_a(2)\)[/tex].
Now we combine all the simplified expressions:
[tex]\[
\begin{align*}
\log_a(\sqrt{2}) + \log_a(8) + \log_a\left(\frac{1}{2}\right) &= \frac{1}{2} \log_a(2) + 3 \log_a(2) - \log_a(2)\\
&= \left(\frac{1}{2} + 3 - 1\right) \log_a(2)\\
&= \left(\frac{1}{2} + 2\right) \log_a(2)\\
&= \left(\frac{1}{2} + \frac{4}{2}\right) \log_a(2)\\
&= \left(\frac{5}{2}\right) \log_a(2).
\end{align*}
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\frac{5}{2} \log_a(2).
\][/tex]
Therefore, in the form [tex]\(p \log_a(2)\)[/tex], we have [tex]\(p = \frac{5}{2}\)[/tex].
