Let's solve the given equation step-by-step:
The equation given is
[tex]\[ \log_2(3x - 7) = 3. \][/tex]
To solve for [tex]\(x\)[/tex], we can convert the logarithmic equation into an exponential equation. Recall the property that [tex]\(\log_b(a) = c\)[/tex] implies [tex]\(a = b^c\)[/tex]. In this case, [tex]\(b = 2\)[/tex], [tex]\(a = 3x - 7\)[/tex], and [tex]\(c = 3\)[/tex]. Therefore, we can rewrite the equation as:
[tex]\[ 3x - 7 = 2^3. \][/tex]
We know that:
[tex]\[ 2^3 = 8. \][/tex]
So the equation becomes:
[tex]\[ 3x - 7 = 8. \][/tex]
Next, we solve for [tex]\(x\)[/tex]. First, add 7 to both sides of the equation to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ 3x - 7 + 7 = 8 + 7, \][/tex]
which simplifies to:
[tex]\[ 3x = 15. \][/tex]
Now, divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{15}{3}. \][/tex]
This simplifies to:
[tex]\[ x = 5. \][/tex]
So the solution to the equation [tex]\(\log_2(3x - 7) = 3\)[/tex] is:
[tex]\[ x = 5. \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{5}. \][/tex]
