To determine which of the given options is equivalent to the logarithmic equation [tex]\( x = \ln 13 \)[/tex], let's start by understanding what the expression [tex]\( x = \ln 13 \)[/tex] means.
The function [tex]\(\ln\)[/tex] is the natural logarithm, which is the logarithm to the base [tex]\(e\)[/tex], where [tex]\(e\)[/tex] is an irrational constant approximately equal to 2.71828. By definition, if [tex]\( x = \ln 13 \)[/tex], then [tex]\( e^x = 13 \)[/tex].
This can be interpreted as follows:
- The natural logarithm [tex]\( \ln 13 \)[/tex] is the power to which [tex]\(e\)[/tex] must be raised to get 13. So if we raise [tex]\(e\)[/tex] to the power of [tex]\(x\)[/tex], we should get 13.
Now, let's match this understanding with the given options:
A. [tex]\( x = \log_{10} 13 \)[/tex]:
- This option refers to the logarithm of 13 with base 10, not base [tex]\( e\)[/tex]. Therefore, this is not equivalent to [tex]\( x = \ln 13 \)[/tex].
B. [tex]\( x^{13} = e \)[/tex]:
- This suggests that [tex]\(x\)[/tex] raised to the 13th power equals [tex]\(e\)[/tex], which does not align with our interpretation [tex]\( x = \ln 13 \)[/tex].
C. [tex]\( e^x = 13 \)[/tex]:
- This equation directly represents the exponential form of the natural logarithm. If [tex]\( x = \ln 13 \)[/tex], then it must be true that [tex]\( e^x = 13 \)[/tex]. This is the correct equivalent equation.
D. [tex]\( e^{13} = x \)[/tex]:
- This implies that [tex]\( e \)[/tex] raised to the power of 13 equals [tex]\(x\)[/tex], which is incorrect because [tex]\( x = \ln 13 \)[/tex], not [tex]\(e^{13}\)[/tex].
Given these explanations, the correct multiple-choice option that is equivalent to [tex]\( x = \ln 13 \)[/tex] is:
C. [tex]\( e^x = 13 \)[/tex]
