To solve the problem
[tex]\[
\log_2(4 \sqrt[3]{2}),
\][/tex]
we will proceed step-by-step.
First, we need to rewrite the expression inside the logarithm in a simpler form. Given the expression [tex]\(4 \sqrt[3]{2}\)[/tex], we can express it using properties of exponents:
1. Simplify [tex]\( 4 \sqrt[3]{2} \)[/tex]
We know that:
[tex]\[
4 = 2^2
\][/tex]
and writing [tex]\(\sqrt[3]{2}\)[/tex] in exponential form:
[tex]\[
\sqrt[3]{2} = 2^{1/3}
\][/tex]
Therefore, the expression inside the logarithm becomes:
[tex]\[
4 \sqrt[3]{2} = 2^2 \cdot 2^{1/3}
\][/tex]
Using the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we can combine the exponents:
[tex]\[
2^2 \cdot 2^{1/3} = 2^{2 + \frac{1}{3}}
\][/tex]
Rewrite the exponent [tex]\(2 + \frac{1}{3}\)[/tex] as a single fraction:
[tex]\[
2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}
\][/tex]
Therefore:
[tex]\[
4 \sqrt[3]{2} = 2^{7/3}
\][/tex]
2. Calculate the logarithm
Now we need to find:
[tex]\[
\log_2(2^{7/3})
\][/tex]
Using the property of logarithms that states [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex] and recognizing that [tex]\(\log_2 (2) = 1\)[/tex], we can simplify:
[tex]\[
\log_2 (2^{7/3}) = \frac{7}{3} \log_2 (2) = \frac{7}{3} \cdot 1 = \frac{7}{3}
\][/tex]
This means our result is:
[tex]\[
\log_2 (4 \sqrt[3]{2}) = \frac{7}{3}
\][/tex]
Converting the fraction [tex]\(\frac{7}{3}\)[/tex] to a decimal form gives:
[tex]\[
\frac{7}{3} \approx 2.3333333333333335
\][/tex]
Thus, the value of [tex]\(\log_2(4 \sqrt[3]{2})\)[/tex] is approximately:
[tex]\[
2.3333333333333335
\][/tex]
