To solve the logarithmic inequality [tex]\(\log_3(x + 1) \leq 2\)[/tex], we will proceed step-by-step:
1. Understanding the Inequality: We need to find the values of [tex]\(x\)[/tex] such that the logarithm to base 3 of [tex]\(x + 1\)[/tex] is less than or equal to 2.
2. Rewrite the Inequality: First, we rewrite the logarithmic inequality in the exponential form.
[tex]\[
\log_3(x + 1) \leq 2
\][/tex]
Recall the property of logarithms: [tex]\(\log_b(a) = c\)[/tex] is equivalent to [tex]\(b^c = a\)[/tex]. So,
[tex]\[
\log_3(x + 1) \leq 2 \implies x + 1 \leq 3^2
\][/tex]
3. Calculate the Exponentiation:
[tex]\[
3^2 = 9
\][/tex]
Thus,
[tex]\[
x + 1 \leq 9
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
[tex]\[
x \leq 9 - 1
\][/tex]
[tex]\[
x \leq 8
\][/tex]
5. Domain Consideration: Since the argument of a logarithm must be positive,
[tex]\[
x + 1 > 0
\][/tex]
[tex]\[
x > -1
\][/tex]
6. Combine the Results: The values of [tex]\(x\)[/tex] must satisfy both conditions:
[tex]\[
-1 < x \leq 8
\][/tex]
Therefore, the correct solution is:
[tex]\[
\boxed{-1 < x \leq 8}
\][/tex]
Thus, the correct answer from the given choices is:
[tex]\[ \text{-1
