Answer :
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
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
