Answer :

Sure, let's break down the given inequality and solve it step-by-step. The inequality we have is:

[tex]\[ \log_2(x + 3) - \log_2(x) + \log_2(2x^2) \leq 3 \][/tex]

First, use the properties of logarithms to simplify the expression.

Step 1: Combine the logarithmic terms.

We apply the properties of logarithms:
- [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex]
- [tex]\(\log_b(a) + \log_b(b) = \log_b(ab)\)[/tex]

Applying these properties:

[tex]\[ \log_2(x+3) - \log_2(x) = \log_2\left(\frac{x+3}{x}\right) \][/tex]

[tex]\[ \log_2(2x^2) = \log_2(2) + \log_2(x^2) = 1 + 2\log_2(x) \][/tex]

Combining all the logarithmic terms gives us:

[tex]\[ \log_2\left(\frac{x+3}{x}\right) + 1 + 2\log_2(x) \leq 3 \][/tex]

Step 2: Further simplify the inequality.

Substitute back into the inequality:

[tex]\[ \log_2\left(\frac{x+3}{x}\right) + 1 + 2\log_2(x) \leq 3 \][/tex]

Combine the constant terms:

[tex]\[ \log_2\left(\frac{x+3}{x}\right) + 2\log_2(x) \leq 2 \][/tex]

Step 3: Combine the logarithms under a single logarithm term.

Use the property:

[tex]\(\log_b(a) + \log_b(b) = \log_b(ab)\)[/tex]

[tex]\[ \log_2\left(\frac{x+3}{x} \cdot x^2\right) \leq 2 \][/tex]

Simplify the inside of the logarithm:

[tex]\[ \log_2\left(\frac{x+3}{x} \cdot x^2\right) = \log_2(x^2 + 3x) \][/tex]

Step 4: Exponentiate both sides to remove the logarithm term.

We have:

[tex]\[ \log_2(x^2 + 3x) \leq 2 \][/tex]

By exponentiating both sides with base 2, we get:

[tex]\[ x^2 + 3x \leq 2^2 \][/tex]

Simplify the exponentiation:

[tex]\[ x^2 + 3x \leq 4 \][/tex]

Step 5: Solve the quadratic inequality.

Rewrite the inequality in standard quadratic form:

[tex]\[ x^2 + 3x - 4 \leq 0 \][/tex]

Factor the quadratic equation:

[tex]\[ (x + 4)(x - 1) \leq 0 \][/tex]

Step 6: Solve the factored inequality.

The critical points are [tex]\(x = -4\)[/tex] and [tex]\(x = 1\)[/tex]. These points divide the number line into three intervals: [tex]\( (-\infty, -4) \)[/tex], [tex]\( (-4, 1) \)[/tex], and [tex]\( (1, \infty) \)[/tex].

Determine where the quadratic expression is non-positive by testing values from each interval:

1. [tex]\( x < -4\)[/tex] (e.g., [tex]\( x = -5 \)[/tex]):
[tex]\[ (-5 + 4)(-5 - 1) = (-1)(-6) = 6 \quad (\text{positive}) \][/tex]

2. [tex]\(-4 \leq x \leq 1 \)[/tex]:
[tex]\[ (x + 4)(x - 1) \leq 0 \quad (\text{interval satisfies the inequality}) \][/tex]

3. [tex]\( x > 1 \)[/tex] (e.g., [tex]\( x = 2 \)[/tex]):
[tex]\[ (2 + 4)(2 - 1) = 6 \quad (\text{positive}) \][/tex]

Thus, the inequality [tex]\( (x + 4)(x - 1) \leq 0 \)[/tex] holds for:

[tex]\[-4 \leq x \leq 1\][/tex]

Step 7: Verify the domain constraints.

The original logarithmic expressions impose constraints:
[tex]\[ x + 3 > 0 \quad \text{and} \quad x > 0 \implies x > 0 \][/tex]

Thus, [tex]\(x > 0\)[/tex] adds an additional constraint, limiting the solution to:

[tex]\[0 < x \leq 1\][/tex]

Final Solution:

The solution to the inequality [tex]\(\log_2(x + 3) - \log_2(x) + \log_2(2x^2) \leq 3\)[/tex] is:

[tex]\[ x \in (0, 1] \][/tex]