To simplify the expression [tex]\( 2(\ln 3 - \ln 5) \)[/tex] and write it as a single natural logarithm, let's use the properties of logarithms step-by-step.
1. Apply the Power Rule of Logarithms:
The power rule states that [tex]\( a \cdot \ln b = \ln(b^a) \)[/tex]. So, for the given expression [tex]\( 2(\ln 3 - \ln 5) \)[/tex], we distribute the 2 to each term inside the parentheses:
[tex]\[
2(\ln 3) - 2(\ln 5)
\][/tex]
Using the power rule, this becomes:
[tex]\[
\ln(3^2) - \ln(5^2)
\][/tex]
Simplifying further, we get:
[tex]\[
\ln 9 - \ln 25
\][/tex]
2. Apply the Quotient Rule of Logarithms:
The quotient rule states that [tex]\( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \)[/tex]. Applying this rule to our expression, we get:
[tex]\[
\ln \left( \frac{9}{25} \right)
\][/tex]
Therefore, the expression [tex]\( 2(\ln 3 - \ln 5) \)[/tex] simplified as a single natural logarithm is:
[tex]\[
\boxed{\ln \left( \frac{9}{25} \right)}
\][/tex]
So, the correct answer is [tex]\(\ln \left( \frac{9}{25} \right)\)[/tex].
