To find the condensed expression for [tex]\(2 \ln x + \ln 3\)[/tex], we can use properties of logarithms.
1. Power Rule of Logarithms: The expression [tex]\(2 \ln x\)[/tex] can be rewritten using the power rule, which states that [tex]\(a \ln b = \ln b^a\)[/tex]. Thus,
[tex]\[
2 \ln x = \ln x^2.
\][/tex]
2. Sum Rule of Logarithms: Next, we apply the sum rule for logarithms, which states that [tex]\(\ln a + \ln b = \ln (a \cdot b)\)[/tex]. So, we add [tex]\(\ln x^2\)[/tex] and [tex]\(\ln 3\)[/tex]:
[tex]\[
\ln x^2 + \ln 3 = \ln (x^2 \cdot 3).
\][/tex]
3. Combine the Logarithms: Combining them, the expression becomes:
[tex]\[
\ln (3 x^2).
\][/tex]
Therefore, the condensed expression for [tex]\(2 \ln x + \ln 3\)[/tex] is [tex]\(\ln (3 x^2)\)[/tex].
Answer: [tex]\(\ln (3 x^2)\)[/tex]
