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Simplify each expression.

1. [tex]\(\ln e^3 = \square\)[/tex]
2. [tex]\(\ln e^{2y} = \square\)[/tex]

Sagot :

GinnyAnswer
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]
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