To evaluate [tex]\( e^{x \ln y} \)[/tex] when [tex]\( x = 3 \)[/tex] and [tex]\( y = 2 \)[/tex], follow these steps:
1. Substitute the given values:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = 2 \][/tex]
2. Express the problem with the substituted values:
[tex]\[ e^{3 \ln 2} \][/tex]
3. Understand the expression:
- The expression involves the natural logarithm (ln) and the exponential function (e).
- The natural logarithm of 2, denoted as [tex]\( \ln 2 \)[/tex], is the power to which the base [tex]\( e \)[/tex] must be raised to produce the number 2.
4. Simplify the exponent:
[tex]\[ 3 \ln 2 \][/tex]
This represents three times the natural logarithm of 2.
5. Exponentiate the result:
- The key identity to use here is [tex]\( e^{\ln k} = k \)[/tex] for any positive [tex]\( k \)[/tex].
6. Combine the exponentiation with the logarithm:
[tex]\[ e^{3 \ln 2} = (e^{\ln 2})^3 \][/tex]
7. Simplify the inner exponential term:
- Using the identity [tex]\( e^{\ln 2} = 2 \)[/tex]:
[tex]\[ (e^{\ln 2})^3 = 2^3 \][/tex]
8. Calculate the final value:
[tex]\[ 2^3 = 8 \][/tex]
Therefore, the value of [tex]\( e^{x \ln y} \)[/tex] when [tex]\( x = 3 \)[/tex] and [tex]\( y = 2 \)[/tex] is:
[tex]\[ e^{3 \ln 2} = 8 \][/tex]
(Note: Given the numerical result shared earlier, the actual computed value was [tex]\( 7.999999999999998 \)[/tex], which is effectively 8 within a tiny margin of error due to how floating-point arithmetic works in computational scenarios.)
