Question

which expressions are equivalent to \\(2 \ln a + 2 \ln b - \ln a\\)?

check all that apply.

• \\(\ln ab^2 - \ln a\\)
• \\(\ln a + 2 \ln b\\)
• \\(\ln a^2 + \ln b^2 - \ln a\\)
• \\(2 \ln ab\\)
• \\(\ln ab^2\\)

Explanation:

⚡ Using what you learned: properties of logarithms

Step 1: Combine like terms

$$ 2 \ln a + 2 \ln b - \ln a = (2 \ln a - \ln a) + 2 \ln b = \ln a + 2 \ln b $$

This matches the second option: \(\ln a + 2 \ln b\).

Step 2: Apply the power property of logarithms

Using \(n \ln x = \ln(x^n)\):

$$ 2 \ln a + 2 \ln b - \ln a = \ln(a^2) + \ln(b^2) - \ln a $$

This matches the third option: \(\ln a^2 + \ln b^2 - \ln a\).

Step 3: Simplify further using product and quotient properties

Using \(\ln a + 2 \ln b\):

$$ \ln a + \ln(b^2) = \ln(a \cdot b^2) = \ln(ab^2) $$

This matches the fifth option: \(\ln ab^2\).

Answer:

The equivalent expressions are:

• \(\ln a + 2 \ln b\) (Option 2)
• \(\ln a^2 + \ln b^2 - \ln a\) (Option 3)
• \(\ln ab^2\) (Option 5)