Let's go through both parts step-by-step:

1. Rewrite $e^2 = y$ as a logarithmic equation:

To convert the exponential equation $e^2 = y$ to a logarithmic form, you can use the general relationship between exponential and logarithmic functions:

$a^b = c \quad \text{can be rewritten as} \quad \log_a(c) = b$

For your equation $e^2 = y$, where the base $e$ is Euler's number, we apply the rule to get:

$\ln(y) = 2$

Where $\ln$ represents the natural logarithm (logarithm with base $e$).

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2. Rewrite $\ln(x) = 6$ as an exponential equation:

To convert the logarithmic equation $\ln(x) = 6$ into its exponential form, recall that:

$\ln(x) = y \quad \text{is equivalent to} \quad e^y = x$

So for your equation $\ln(x) = 6$, we rewrite it as:

$x = e^6$

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Summary:

1. $e^2 = y$ becomes $\ln(y) = 2$.
2. $\ln(x) = 6$ becomes $x = e^6$.

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Would you like more details or any further explanations on logarithmic and exponential equations?

Here are some additional questions for exploration:

1. How do natural logarithms differ from logarithms with other bases?
2. Can you solve $\ln(x) = 6$ numerically?
3. What is the inverse of an exponential function like $e^x$?
4. How does the change of base formula for logarithms work?
5. What happens to the graph of $e^x$ and $\ln(x)$?

Tip: Remember that natural logarithms are especially useful in calculus, especially for integration and differentiation involving exponential growth or decay.