Let's solve the expression [tex]\(\log_{100 \sqrt{64}} \left(2^3 \times 512^{0.1}\right)\)[/tex] step-by-step.
1. Simplify the expression inside the logarithm:
[tex]\[
2^3 \times 512^{0.1}
\][/tex]
Firstly, notice that [tex]\(512\)[/tex] can be rewritten as a power of 2:
[tex]\[
512 = 2^9
\][/tex]
Therefore,
[tex]\[
512^{0.1} = (2^9)^{0.1} = 2^{9 \times 0.1} = 2^{0.9}
\][/tex]
Now, substituting this back into the original expression:
[tex]\[
2^3 \times 2^{0.9} = 2^{3+0.9} = 2^{3.9}
\][/tex]
2. Simplify the base of the logarithm:
The base is given as [tex]\(100 \sqrt{64}\)[/tex].
We know that [tex]\(\sqrt{64} = 8\)[/tex],
So,
[tex]\[
100 \sqrt{64} = 100 \times 8 = 800
\][/tex]
3. Rewrite the logarithm:
Now, our expression simplifies to:
[tex]\[
\log_{800} \left(2^{3.9}\right)
\][/tex]
4. Using the change of base formula:
The change of base formula for logarithms is:
[tex]\[
\log_b(a) = \frac{\log(a)}{\log(b)}
\][/tex]
Here, [tex]\(a = 2^{3.9}\)[/tex] and [tex]\(b = 800\)[/tex].
So,
[tex]\[
\log_{800} \left(2^{3.9}\right) = \frac{\log(2^{3.9})}{\log(800)}
\][/tex]
5. Simplifying the numerator [tex]\(\log(2^{3.9})\)[/tex]:
We know from the properties of logarithms:
[tex]\[
\log(2^{3.9}) = 3.9 \cdot \log(2)
\][/tex]
6. Calculate the value inside the logarithm:
The value of [tex]\(2^{3.9}\)[/tex] is approximately 14.929:
[tex]\[
2^{3.9} \approx 14.93
\][/tex]
7. Compute the logarithms:
Given the result from calculations:
[tex]\[
\log(14.929) \approx 1.174
\][/tex]
[tex]\[
\log(800) \approx 2.905
\][/tex]
8. Combine the results together:
Finally:
[tex]\[
\frac{\log(14.929)}{\log(800)} = \frac{1.174}{2.905} \approx 0.404
\][/tex]
So, the value of [tex]\(\log_{800} \left(2^{3.9}\right)\)[/tex] is approximately 0.404.
Hence,
[tex]\[
\log_{100 \sqrt{64}}\left(2^3 \times 512^{0.1}\right) \approx 0.404
\][/tex]
Therefore, the value of the given logarithmic expression is approximately [tex]\(0.404\)[/tex].
