To determine which of the given logarithmic equations is equivalent to the exponential equation [tex]\( e^x = 31 \)[/tex], we need to use the properties of logarithms, specifically the natural logarithm, as it relates to the exponential function.
1. The given equation is:
[tex]\[
e^x = 31
\][/tex]
2. To convert this exponential equation into a logarithmic equation, take the natural logarithm (denoted as [tex]\(\ln\)[/tex]) of both sides of the equation. The natural logarithm is the inverse operation of exponentiation with base [tex]\(e\)[/tex].
Applying the natural logarithm:
[tex]\[
\ln(e^x) = \ln(31)
\][/tex]
3. We utilize the property of logarithms that [tex]\(\ln(e^x) = x \cdot \ln(e)\)[/tex]. Also, knowing that [tex]\(\ln(e) = 1\)[/tex], we can simplify the left side of the equation:
[tex]\[
x \cdot \ln(e) = \ln(31)
\][/tex]
[tex]\[
x \cdot 1 = \ln(31)
\][/tex]
[tex]\[
x = \ln(31)
\][/tex]
Therefore, the logarithmic equation equivalent to the given exponential equation [tex]\( e^x = 31 \)[/tex] is:
[tex]\[
\ln(31) = x
\][/tex]
This corresponds to option:
c. [tex]\(\ln 31 = x\)[/tex]
