Question

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:

⚡ Using what you learned: properties of logarithms

Step 1: Apply the Power Property

Apply the power property of logarithms, \( a \ln b = \ln(b^a) \), to the term \( 2 \ln 5 \):

$$ 2 \ln 5 = \ln(5^2) = \ln 25 $$

The expression becomes:

$$ \ln x + \ln 25 + \ln 1 $$

Step 2: Apply the Product Property

Apply the product property of logarithms, \( \ln a + \ln b = \ln(ab) \), to combine the terms:

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

This matches the third option: \( \ln 25x + \ln 1 \).

Step 3: Simplify using the Identity Property

Since \( \ln 1 = 0 \), simplify the expression further:

$$ \ln 25x + \ln 1 = \ln 25x + 0 = \ln 25x $$

This matches the fourth option: \( \ln 25x \).

Answer:

The equivalent expressions are:

• \( \ln 25x + \ln 1 \)
• \( \ln 25x \)