The graph shown is on a semi-logarithmic scale, where the $y$-axis represents $\log(y)$ and the $x$-axis is linear. This setup typically means that the plotted relationship is exponential in nature.

Steps to Find the Equation

1. Identify the Equation Format: On a semi-log plot, a straight line indicates an exponential function. Since the y-axis is in terms of $\log(y)$, the equation we expect for $y(x)$ is: $y = a \cdot e^{bx}$ where $a$ is the initial value, and $b$ is the rate of growth (or decay if $b < 0$).

2. Analyze the Line: Since the graph of $\log(y)$ is linear with respect to $x$, we can model $\log(y) = bx + \log(a)$.

3. Determine Slope and Intercept:

   • By examining two points on the line, determine the slope, $b$.
   • Identify the intercept on the $\log(y)$-axis, which gives $\log(a)$.

4. Convert to Exponential Form: After calculating $a$ and $b$, use these values in the form $y = a \cdot e^{bx}$.

Let me know if you’d like help finding specific points on the line for a more detailed calculation!

Questions to Consider:

1. How does a semi-logarithmic scale help in identifying exponential relationships?
2. Why does a straight line on a semi-log plot represent exponential growth or decay?
3. What role does the slope of the line on a $\log(y)$ vs. $x$ plot play in determining the rate of growth?
4. How would the equation change if the line had a negative slope?
5. What are some real-world applications of exponential trends represented on semi-logarithmic graphs?

Tip: In semi-logarithmic plots, remember that the slope corresponds to the exponent's coefficient in the exponential equation.