https://file.aiquickdraw.com/mathbot/file/1730244065267vx74e16o.jpg

To find the equation for the given graph on a semi-logarithmic scale, let's analyze the information.

### Understanding the Semi-Log Scale
The \( y \)-axis is labeled as \( \log(y) \), indicating that this is a semi-logarithmic plot, where:
- The \( x \)-axis is in a linear scale.
- The \( y \)-axis represents \( \log(y) \), meaning each unit on the \( y \)-axis corresponds to an exponential increase in \( y \).

This suggests the function is of the form:
\[
y = a \cdot b^x
\]
where \( \log(y) = \log(a) + x \cdot \log(b) \), and thus, the graph of \( \log(y) \) versus \( x \) should be a straight line.

### Analyzing the Line
On the graph:
1. Identify two points on the line, as we need them to determine the slope and intercept.
   - Looking at the image, the line seems to pass through points approximately at \( (0, -3) \) and \( (2, -2) \).

2. Using these points, calculate the slope \( m \):
   \[
   m = \frac{-2 - (-3)}{2 - 0} = \frac{1}{2}
   \]

3. The equation of the line in terms of \( \log(y) \) is:
   \[
   \log(y) = \frac{1}{2}x - 3
   \]

### Converting to Exponential Form
Now, rewrite this in terms of \( y \):
1. Recall that \( \log(y) = \frac{1}{2}x - 3 \).
2. Converting from logarithmic to exponential form:
   \[
   y = 10^{\frac{1}{2}x - 3}
   \]
3. Simplify:
   \[
   y = 10^{-3} \cdot 10^{\frac{1}{2}x} = \left(10^{-3}\right) \cdot \left(10^{\frac{1}{2}}\right)^x
   \]
4. Thus, \( a = 10^{-3} \) and \( b = 10^{\frac{1}{2}} \).

So, the final equation is:
\[
y(x) = 0.001 \cdot (10^{0.5})^x
\]

Let me know if you'd like more details or if you have questions about any part of the solution!

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Here are 5 related questions to expand your understanding:
1. How does a semi-log scale differ from a log-log scale?
2. What types of real-world data are best represented by exponential functions?
3. How would the slope change if the function’s growth rate increased?
4. Why is the y-axis in log scale rather than linear scale for this graph?
5. How do we interpret the intercept in an exponential function on a semi-log scale?

**Tip:** When working with semi-log scales, keep in mind that the line's slope represents the growth rate of the exponential function.

Find an equation for the graph shown on a semi-logarithmic scale, where the y-axis represents log(y) and the graph displays a straight line.

Learn to find the exponential equation from a semi-logarithmic graph where the y-axis represents log(y). This problem covers concepts in exponential functions, semi-log scale interpretation, and logarithmic transformations. Suitable for students in Grades 10-12.

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