To determine which expressions are equivalent to [tex]\(\ln \left(e^2\right)\)[/tex], we can use the properties of logarithms. Specifically, we can use the power rule for logarithms which states:
[tex]\[
\ln \left(a^b\right) = b \cdot \ln(a)
\][/tex]
Applying this rule to [tex]\(\ln \left(e^2\right)\)[/tex]:
[tex]\[
\ln \left(e^2\right) = 2 \cdot \ln(e)
\][/tex]
We know that [tex]\(\ln(e)\)[/tex] is a special natural logarithm value:
[tex]\[
\ln(e) = 1
\][/tex]
Therefore, our expression becomes:
[tex]\[
2 \cdot \ln(e) = 2 \cdot 1 = 2
\][/tex]
With this, we can see which of the provided options are equivalent to [tex]\(\ln \left(e^2\right)\)[/tex]:
A. 1 — This is not equivalent since we found that [tex]\(\ln \left(e^2\right)\)[/tex] equals 2.
B. 2 — This is correct because we determined that [tex]\(\ln \left(e^2\right)\)[/tex] simplifies to 2.
C. [tex]\(2e\)[/tex] — This is not equivalent since adding [tex]\(e\)[/tex] introduces an additional factor that changes the value.
D. [tex]\(2 \cdot \ln(e)\)[/tex] — This is correct because this matches our intermediate expression before substituting in the value of [tex]\(\ln(e)\)[/tex].
Thus, the equivalent expressions to [tex]\(\ln \left(e^2\right)\)[/tex] are:
- B. 2
- D. [tex]\(2 \cdot \ln e\)[/tex]
