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\\)

which expressions are equivalent to \\(\ln x + 2 \ln 5 + \ln 1\\)?

check all that apply.

• \\(2 \ln 5x\\)
• \\(\ln (x + 26)\\)
• \\(\ln 25x + \ln 1\\)
• \\(\ln 25x\\)

Explanation:

Simplify the first expression

Using the Logarithmic Power Rule and Logarithmic Product Rule knowledge points

$$
LATEXBLOCK0
$$

Also, rewriting the original expression using the power rule directly:

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

Evaluate the first set of options

Using the Logarithmic Properties knowledge point

• \(\ln ab^2 - \ln a\): Incorrect.
• \(\ln a + 2\ln b\): Correct (matches step 1).
• \(\ln a^2 + \ln b^2 - \ln a\): Correct (matches step 1).
• \(2\ln ab\): Incorrect, as \(2\ln ab = \ln(ab)^2 = \ln(a^2b^2)\).
• \(\ln ab^2\): Correct (matches step 1).

Simplify the second expression

Using the Logarithmic Power Rule, Logarithmic Product Rule, and Logarithmic Evaluation knowledge points

$$
LATEXBLOCK1
$$

Alternatively, keeping the \(\ln 1\) term:

$$ \ln x + 2\ln 5 + \ln 1 = \ln 25x + \ln 1 $$

Evaluate the second set of options

Using the Logarithmic Properties knowledge point

• \(2\ln 5x\): Incorrect, as \(2\ln 5x = \ln(25x^2)\).
• \(\ln(x + 26)\): Incorrect.
• \(\ln 25x + \ln 1\): Correct (since \(\ln 1 = 0\)).
• \(\ln 25x\): Correct (matches step 3).

Answer:

Question 1

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

Question 2

• [ ] \(2\ln 5x\)
• [ ] \(\ln(x + 26)\)
• [x] \(\ln 25x + \ln 1\) (Correct answer)
• [x] \(\ln 25x\) (Correct answer)