3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)

Verified step by step guidance

1. Recall the logarithm property that states: \(\ln(a) + \ln(b) = \ln(ab)\). This allows us to combine the sum of logarithms into a single logarithm of the product.
2. Identify the two expressions inside the logarithms: the first is \(3x^{2} - 9x\) and the second is \(\frac{1}{3}x\).
3. Multiply the two expressions inside the logarithms: \((3x^{2} - 9x) \times \left(\frac{1}{3}x\right)\).
4. Simplify the product by distributing and combining like terms: multiply \$3x^{2}\( by \(\frac{1}{3}x\) and \)-9x$ by \(\frac{1}{3}x\).
5. Write the final expression as a single logarithm: \(\ln\left(\text{simplified product}\right)\).

Key Concepts

Properties of Logarithms

Logarithmic properties allow the simplification and combination of logarithmic expressions. Key properties include the product rule (ln a + ln b = ln(ab)), the quotient rule (ln a - ln b = ln(a/b)), and the power rule (ln(a^b) = b ln a). These rules help rewrite multiple logarithmic terms as a single logarithm.

Domain of Logarithmic Functions

The domain of a logarithmic function consists of all positive real numbers inside the logarithm. When combining logarithms, it is essential to ensure the resulting argument remains positive, as ln(x) is undefined for x ≤ 0. This affects the validity of the simplified expression.

Algebraic Simplification

Algebraic simplification involves factoring and reducing expressions inside the logarithms before applying logarithmic properties. For example, factoring 3x² - 9x as 3x(x - 3) can make it easier to combine terms and simplify the overall expression.