https://file.aiquickdraw.com/mathbot/file/1727998713479o8n0n28d.jpg

From the image, we see a graph of the natural logarithmic function \( g(x) = \ln(x) \). The point marked on the graph has the coordinates \( (X = 2.76, Y = 1.0152307) \). The task is to write the logarithmic equation associated with the displayed point.

The general form of the logarithmic equation is:

\[
Y = \ln(X)
\]

Using the point provided \( (X = 2.76, Y = 1.0152307) \), we can directly substitute the \( X \)-value into the equation:

\[
Y = \ln(2.76)
\]

We can verify the logarithmic equation by checking if \( \ln(2.76) \) equals the given \( Y \)-value \( 1.0152307 \). Indeed, this is approximately true since:

\[
\ln(2.76) \approx 1.0152307
\]

Thus, the logarithmic equation is:

\[
Y = \ln(2.76)
\]

Would you like to go over the exponential counterpart of this equation or any other details?

Here are 5 related questions for further exploration:
1. What is the inverse of the logarithmic function \( Y = \ln(X) \)?
2. How do we rewrite the equation \( Y = \ln(X) \) in exponential form?
3. What is the derivative of \( g(x) = \ln(x) \)?
4. How do the properties of logarithms apply when solving equations like \( \ln(a \cdot b) \)?
5. What happens to \( \ln(x) \) as \( x \to 0^+ \) or \( x \to \infty \)?

**Tip:** The natural logarithm \( \ln(x) \) is defined only for positive \( x \)-values, which is why the graph only appears for \( x > 0 \).

Write the logarithmic and exponential equations associated with the given graph of g(x) = ln(x), and identify the logarithmic equation for a point with coordinates (X = 2.76, Y = 1.0152307).

Discover how to write logarithmic and exponential equations based on the graph of g(x) = ln(x). Learn how to interpret and verify the logarithmic equation using a point with coordinates (X = 2.76, Y = 1.0152307), suitable for high school students exploring advanced functions.

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