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 terms:

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

Evaluate the first set of choices

Using the Logarithmic Power Rule and Logarithmic Product Rule knowledge points

• \(\ln ab^2 - \ln a\): Incorrect, as \(\ln(ab^2) - \ln a = \ln b^2 = 2\ln b\).
• \(\ln a + 2\ln b\): Correct, matches the simplified form.
• \(\ln a^2 + \ln b^2 - \ln a\): Correct, matches the power rule expansion.
• \(2\ln ab\): Incorrect, as \(2\ln ab = \ln(ab)^2 = \ln(a^2b^2)\).
• \(\ln ab^2\): Correct, matches the fully simplified single logarithm.

Simplify the second expression

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

$$
LATEXBLOCK1
$$

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

$$ \ln x + \ln 25 + \ln 1 = \ln(25x) + \ln 1 $$

Evaluate the second set of choices

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

• \(2\ln 5x\): Incorrect, as \(2\ln 5x = \ln(25x^2)\).
• \(\ln(x + 26)\): Incorrect, as logarithms do not distribute over addition.
• \(\ln 25x + \ln 1\): Correct, since \(\ln 1 = 0\), this is equivalent to \(\ln 25x\).
• \(\ln 25x\): Correct, matches the fully simplified single logarithm.

Answer:

Question 1

• (A) \(\ln ab^2 - \ln a\)
• (B) \(\ln a + 2\ln b\) (Correct answer)
• (C) \(\ln a^2 + \ln b^2 - \ln a\) (Correct answer)
• (D) \(2\ln ab\)
• (E) \(\ln ab^2\) (Correct answer)

Question 2

• (A) \(2\ln 5x\)
• (B) \(\ln(x + 26)\)
• (C) \(\ln 25x + \ln 1\) (Correct answer)
• (D) \(\ln 25x\) (Correct answer)