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Simplify [tex]\log _a \sqrt{2} + \log _a 8 + \log _a \left( \frac{1}{2} \right)[/tex], giving your answer in the form [tex]p \log _a 2[/tex], where [tex]p[/tex] is a constant.

Sagot :

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].
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