Converting between natural logarithmic and exponential equations Rewrite each equation as requested. (a) Rewrite as a logarithmic equation. $$ e^{y}=9 $$ (b) Rewrite as an exponential equation. $$ \ln x=6 $$ (a) $$ \square $$ (b) $$ \square $$ \begin{tabular}{|l|l|}\hline$$ \frac{\square}{\square} $$ & $$ \square=\square $$ \ \hline\end{tabular}

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Let's rewrite the equations step by step without using the tool calls, as they were not applicable for this conversion.

### (a) Rewrite as a logarithmic equation.
The equation given is:
$$ e^{y} = 9 $$

To convert this to a logarithmic form, we use the definition of logarithms, which states that if $$ a^b = c $$, then $$ \log_a(c) = b $$. Here, $$ a = e $$, $$ b = y $$, and $$ c = 9 $$.

Thus, we can rewrite the equation as:
$$ y = \ln(9) $$

### (b) Rewrite as an exponential equation.
The equation given is:
$$ \ln x = 6 $$

To convert this to an exponential form, we again use the definition of logarithms. Here, $$ a = e $$, $$ b = 6 $$, and $$ c = x $$.

Thus, we can rewrite the equation as:
$$ x = e^{6} $$

### Final Answers
(a) $$ y = \ln(9) $$

(b) $$ x = e^{6} $$

Now, filling in the table:

$$
\begin{tabular}{|l|l|}
\hline
$$ \frac{y}{\ln(9)} $$ & $$ x = e^{6} $$ \ 
\hline
\end{tabular}
$$
(a) $$ y = \ln(9) $$ (b) $$ x = e^{6} $$ $$ \begin{tabular}{|l|l|} \hline $$ \frac{y}{\ln(9)} $$ & $$ x = e^{6} $$ \ \hline \end{tabular} $$

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