To determine which equation is equivalent to the logarithmic equation
[tex]\[ x = \ln 4 \][/tex]
we need to recall that the natural logarithm function, [tex]\(\ln y\)[/tex], is the power to which the base [tex]\(e\)[/tex] (approximately 2.718) must be raised to yield [tex]\(y\)[/tex]. In other words, the equation
[tex]\[ x = \ln 4 \][/tex]
can be converted to its exponential form.
Given [tex]\(\ln y = x\)[/tex] converts to [tex]\(e^x = y\)[/tex], we can apply this rule to our original equation:
1. Start with the equation:
[tex]\[ x = \ln 4 \][/tex]
2. Recall the definition of the natural logarithm: [tex]\(\ln y = x\)[/tex] means [tex]\(e^x = y\)[/tex].
3. Apply this definition to our equation:
[tex]\[ e^x = 4 \][/tex]
Thus, the equivalent equation to [tex]\( x = \ln 4 \)[/tex] is:
[tex]\[ e^x = 4 \][/tex]
Now let's check the options provided:
A. [tex]\( e^4 = x \)[/tex]
B. [tex]\( e^x = 4 \)[/tex]
C. [tex]\( x^4 = e \)[/tex]
D. [tex]\( x = \log_{10} 4 \)[/tex]
The correct choice, based on our conversion, is:
B. [tex]\( e^x = 4 \)[/tex]
