Let's convert the given expression [tex]\(3 \log_2(x) - \left(\log_2(3) - \log_2(x + 4)\right)\)[/tex] into a single logarithm step by step.
1. Rewrite [tex]\(3 \log_2(x)\)[/tex]:
[tex]\[
3 \log_2(x) = \log_2(x^3)
\][/tex]
Using the power rule of logarithms, [tex]\(a \log_b(x) = \log_b(x^a)\)[/tex].
2. Simplify [tex]\(\log_2(3) - \log_2(x + 4)\)[/tex]:
[tex]\[
\log_2(3) - \log_2(x + 4) = \log_2\left(\frac{3}{x + 4}\right)
\][/tex]
Using the quotient rule of logarithms, [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex].
3. Combine the results from steps 1 and 2:
[tex]\[
3 \log_2(x) - \left(\log_2(3) - \log_2(x + 4)\right) = \log_2(x^3) - \log_2\left(\frac{3}{x + 4}\right)
\][/tex]
4. Combine the logarithms using the quotient rule again:
[tex]\[
\log_2(x^3) - \log_2\left(\frac{3}{x + 4}\right) = \log_2\left(\frac{x^3}{\frac{3}{x + 4}}\right)
\][/tex]
5. Simplify inside the logarithm:
[tex]\[
\frac{x^3}{\frac{3}{x + 4}} = x^3 \cdot \frac{x + 4}{3} = \frac{x^3 (x + 4)}{3}
\][/tex]
6. Write the final combined logarithm:
[tex]\[
\log_2\left( \frac{x^3 (x + 4)}{3} \right)
\][/tex]
Thus, the expression [tex]\(3 \log_2(x) - \left(\log_2(3) - \log_2(x + 4)\right)\)[/tex] written as a single logarithm is:
[tex]\[
\boxed{\log_2\left( \frac{x^3 (x + 4)}{3} \right)}
\][/tex]
So, the correct option from the given multiple choices is:
[tex]\[
\log_2\left[\frac{x^3 (x + 4)}{3}\right]
\][/tex]
Thus, the correct answer is option 1.
