https://file.aiquickdraw.com/mathbot/file/17292227686817bf1mk35.jpg

Give the equation of the natural logarithm graphed in the figure.

The graph you provided appears to be the plot of a natural logarithmic function. Natural logarithmic functions have the general form:

\[
y = \ln(x)
\]

However, based on the graph and the marked point \(\left(\frac{1}{e}, -3\right)\), the equation seems to have been vertically shifted and possibly scaled.

### Observations:
1. The marked point \(\left(\frac{1}{e}, -3\right)\) gives us a clue. The natural logarithm of \(\frac{1}{e}\) is \(-1\), so the function has been transformed.
2. The graph is shifted down by 2 units since \(\ln\left(\frac{1}{e}\right) = -1\) for the regular natural logarithm, but here the \(y\)-value is \(-3\). This indicates a vertical shift by 2 units down.
3. There also appears to be a vertical scaling, as the graph is stretched or compressed. Normally, \(\ln\left(\frac{1}{e}\right) = -1\), but here the corresponding \(y\)-value is \(-3\), suggesting a scaling factor.

### Equation form:
This looks like a transformed logarithmic function of the form:

\[
y = a \ln(x) + b
\]

Using the point \(\left(\frac{1}{e}, -3\right)\), we can substitute to solve for \(a\) and \(b\):
\[
-3 = a \ln\left(\frac{1}{e}\right) + b
\]
Since \(\ln\left(\frac{1}{e}\right) = -1\), this becomes:
\[
-3 = -a + b
\]

Now, to confirm the vertical shift, observe that the graph appears to cross \(y = 0\) at approximately \(x = 1\), implying \(b = 2\) (since \(\ln(1) = 0\)).

Thus, the equation becomes:
\[
y = 2 \ln(x) - 2
\]

This seems to fit the overall behavior of the graph shown in your image.

Would you like further explanation or have any other questions?

### Relative Questions:
1. How does shifting affect the graph of a logarithmic function?
2. What is the significance of the natural logarithm base \(e\) in calculus?
3. How do you determine vertical scaling from a graph of a function?
4. Can you explain the properties of transformations applied to logarithmic functions?
5. What is the domain of the function \(y = 2 \ln(x) - 2\)?

### Tip:
Always check the point where the logarithmic function equals zero, as it helps identify vertical shifts and other transformations in logarithmic graphs.

Learn how to determine the equation of a transformed natural logarithmic function with a vertical shift and scaling based on the given graph. This problem involves analyzing the function y = 2 ln(x) - 2, which includes core concepts such as natural logarithms, graph transformations, and vertical shifts. Suitable for Grades 10-12.

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