1. The problem states that the graph of $\ln y$ against $\ln x$ is a straight line with a point $(7,9)$ and a vertical intercept of 5. 2. For a straight line graph of $\ln y$ versus $\ln x$, the equation is of the form: $$\ln y = m \ln x + c$$ where $m$ is the slope and $c$ is the intercept. 3. Given the intercept $c = 5$, the equation becomes: $$\ln y = m \ln x + 5$$ 4. Using the point $(7,9)$, substitute $\ln x = 7$ and $\ln y = 9$: $$9 = m \times 7 + 5$$ 5. Solve for $m$: $$9 - 5 = 7m$$ $$4 = 7m$$ $$m = \frac{4}{7}$$ 6. Substitute $m$ back into the line equation: $$\ln y = \frac{4}{7} \ln x + 5$$ 7. Rewrite the equation in terms of $y$ by exponentiating both sides: $$y = e^{\ln y} = e^{\frac{4}{7} \ln x + 5} = e^5 \times e^{\frac{4}{7} \ln x}$$ 8. Use the property $e^{a \ln b} = b^a$: $$y = e^5 \times x^{\frac{4}{7}}$$ Final answer: $$y = e^5 x^{\frac{4}{7}}$$