To simplify the given logarithmic expressions, we need to use the fundamental property of logarithms: [tex]\(\ln(e^a) = a\)[/tex].
1. Simplifying [tex]\(\ln(e^3)\)[/tex]:
- The expression [tex]\(\ln(e^3)\)[/tex] involves the natural logarithm [tex]\(\ln\)[/tex] of [tex]\(e\)[/tex] raised to the power of 3.
- According to the property of logarithms: [tex]\(\ln(e^a) = a\)[/tex], we can simplify [tex]\(\ln(e^3)\)[/tex] directly to the exponent, which is 3.
- Therefore, [tex]\(\ln(e^3) = 3\)[/tex].
So, [tex]\[\ln(e^3) = 3.\][/tex]
2. Simplifying [tex]\(\ln(e^{2y})\)[/tex]:
- The expression [tex]\(\ln(e^{2y})\)[/tex] involves the natural logarithm [tex]\(\ln\)[/tex] of [tex]\(e\)[/tex] raised to the power of [tex]\(2y\)[/tex].
- Using the same property of logarithms: [tex]\(\ln(e^a) = a\)[/tex], we can simplify [tex]\(\ln(e^{2y})\)[/tex] directly to the exponent, which is [tex]\(2y\)[/tex].
- Therefore, [tex]\(\ln(e^{2y}) = 2y\)[/tex].
So, [tex]\[\ln(e^{2y}) = 2y.\][/tex]
To summarize:
[tex]\[
\ln(e^3) = 3
\][/tex]
[tex]\[
\ln(e^{2y}) = 2y
\][/tex]
