To solve this problem, we need to combine the logarithmic terms into a single logarithm with a leading coefficient of 1. Here’s a step-by-step solution.

Given Expression

$\frac{1}{2} \log(3x + 2) - 2 \log(x + 1) + \log(x)$

Step 1: Apply the Power Rule

We can apply the power rule of logarithms, which states $a \log(b) = \log(b^a)$, to each term that has a coefficient.

1. For $\frac{1}{2} \log(3x + 2)$: $\frac{1}{2} \log(3x + 2) = \log((3x + 2)^{1/2}) = \log(\sqrt{3x + 2})$

2. For $-2 \log(x + 1)$: $-2 \log(x + 1) = \log((x + 1)^{-2}) = \log\left(\frac{1}{(x + 1)^2}\right)$

Now, rewrite the expression with these transformations: $\log(\sqrt{3x + 2}) + \log(x) + \log\left(\frac{1}{(x + 1)^2}\right)$

Step 2: Apply the Product and Quotient Rules

The expression now has terms that can be combined using the product and quotient rules for logarithms. The product rule states $\log(a) + \log(b) = \log(a \cdot b)$, and the quotient rule states $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$.

Combine the terms: $\log\left(\frac{\sqrt{3x + 2} \cdot x}{(x + 1)^2}\right)$

Final Answer

The simplified expression as a single logarithm is: $\log\left(\frac{x \sqrt{3x + 2}}{(x + 1)^2}\right)$

This is the single logarithmic form with a leading coefficient of 1.

Would you like further details or have any questions?

Related Questions

1. What is the power rule of logarithms and how is it used in simplification?
2. How do the product and quotient rules help in combining logarithmic terms?
3. What steps are necessary to ensure a single logarithmic expression has a leading coefficient of 1?
4. Why is it beneficial to express multiple logarithmic terms as a single logarithm?
5. How do we handle expressions with square roots in logarithmic simplification?

Tip

Always start by applying the power rule to logarithmic terms with coefficients to make simplification easier.