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:

Simplify the given expression

Using the Algebraic Simplification knowledge point

$$
LATEXBLOCK0
$$

Apply logarithmic power rule

Using the Logarithmic Power Rule knowledge point

$$
LATEXBLOCK1
$$

Apply logarithmic product rule

Using the Logarithmic Product Rule knowledge point

$$
LATEXBLOCK2
$$

Evaluate the given options

Using the Logarithmic Product Rule and Logarithmic Power Rule knowledge points

• \(\ln ab^2 - \ln a\): Equivalent to \(\ln a + 2\ln b - \ln a = 2\ln b

e \ln(ab^2)\).

• \(\ln a + 2\ln b\): Matches the simplified form.
• \(\ln a^2 + \ln b^2 - \ln a\): Matches the power rule expansion.
• \(2\ln ab\): Equivalent to \(2\ln a + 2\ln b

e \ln(ab^2)\).

• \(\ln ab^2\): Matches the fully condensed form.

Answer:

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