To convert the given logarithmic equation into its equivalent exponential form, follow these steps:
1. Understand the logarithmic equation provided:
[tex]\[
2 = \ln(x)
\][/tex]
2. Recall the definition of the natural logarithm:
The natural logarithm, [tex]\( \ln(x) \)[/tex], is the power to which the base [tex]\( e \)[/tex] (Euler's number, approximately 2.71828) must be raised to produce the number [tex]\( x \)[/tex]. Specifically, if [tex]\( \ln(x) = 2 \)[/tex], then:
[tex]\[
e^2 = x
\][/tex]
3. By converting from logarithmic form to exponential form, we have:
[tex]\[
x = e^2
\][/tex]
Therefore, the exponential equation that is equivalent to the logarithmic equation [tex]\( 2 = \ln(x) \)[/tex] is:
[tex]\[
x = e^2
\][/tex]
Thus, the correct choice is:
C. [tex]\( x = e^2 \)[/tex]
