To solve the exponential equation [tex]\( e^x = 15.29 \)[/tex] and find its equivalent logarithmic form, follow these steps:
1. Recognize that [tex]\( e^x \)[/tex] is an exponential function, where the base is [tex]\( e \)[/tex], the natural logarithm base. The inverse operation of exponentiation with base [tex]\( e \)[/tex] is taking the natural logarithm (ln).
2. Apply the natural logarithm to both sides of the equation to isolate the variable [tex]\( x \)[/tex]. This uses the property of logarithms that states [tex]\( \ln(e^x) = x \)[/tex] because the natural logarithm and the exponential function are inverses.
[tex]\[
\begin{align*}
\ln(e^x) &= \ln(15.29)
\end{align*}
\][/tex]
3. Use the logarithmic identity [tex]\( \ln(e^x) = x \cdot \ln(e) \)[/tex]. Because [tex]\( \ln(e) = 1 \)[/tex], this simplifies to:
[tex]\[
\begin{align*}
x \cdot \ln(e) &= \ln(15.29) \\
x \cdot 1 &= \ln(15.29) \\
x &= \ln(15.29)
\end{align*}
\][/tex]
Thus, the equivalent logarithmic equation for [tex]\( e^x = 15.29 \)[/tex] is:
[tex]\[
\boxed{\ln(15.29) = x}
\][/tex]
Therefore, the correct option is C. [tex]\(\ln 15.29 = x\)[/tex].
