Question

which expressions are equivalent to \\(\ln x + 2\ln 5 + \ln 1\\)?
choose two correct answers.
\\(\ln 25x + \ln 1\\) \\(2\ln 5x\\) \\(\ln (x + 26)\\) \\(\ln 25x\\)

Explanation:

Step1: Simplify \(2\ln 5\)

Using the logarithm power rule \(a\ln b=\ln(b^a)\), we have \(2\ln 5 = \ln(5^2)=\ln 25\).
So the original expression \(\ln x + 2\ln 5+\ln 1\) becomes \(\ln x+\ln 25+\ln 1\).

Step2: Combine \(\ln x\) and \(\ln 25\)

Using the logarithm product rule \(\ln a+\ln b = \ln(ab)\), we get \(\ln x+\ln 25=\ln(25x)\).
So now the expression is \(\ln(25x)+\ln 1\), which matches the first option.

Step3: Simplify \(\ln 1\)

We know that \(\ln 1 = 0\), so \(\ln(25x)+\ln 1=\ln(25x)+0=\ln(25x)\), which matches the fourth option.

Step4: Analyze other options

• For \(2\ln 5x\): Using the power rule, \(2\ln 5x=\ln((5x)^2)=\ln(25x^2)\), which is not equal to \(\ln x + 2\ln 5+\ln 1=\ln(25x)\).
• For \(\ln(x + 26)\): This is a sum inside the logarithm, not a product, so it's not equivalent to the original expression which involves products (from logarithm properties).

Answer:

A. \(\ln 25x+\ln 1\)
D. \(\ln 25x\) (assuming the fourth option is labeled D, if the original options are labeled as first: A, second: B, third: C, fourth: D)