A project manager at a startup is tracking the growth of the company's subscriber base. The number of active subscribers grew from 100 to 500 over the first 6 months. To predict future growth, the manager uses the exponential model $A = A_0 e^{kt}$, where $k$ represents the monthly growth rate. Arrange the following steps in the correct order to solve for the exact value of the growth rate $k$.

An IT systems administrator is analyzing a sudden exponential increase in server error logs. The number of daily errors grew from 100 to 500 over a 6-hour period. To forecast the number of errors at 24 hours using the continuous exponential growth model $A = A_0 e^{kt}$, the administrator first needs to find the exact growth rate, $k$. Based on the standard exponential growth formula, which equation correctly substitutes the known values to solve for the growth rate?

A business analyst is tracking the growth of a new digital service's subscriber base. Initial data shows that the number of subscribers grew from ${}100$ to ${}500$ over the first 6 weeks of the launch. To forecast future expansion, the analyst applies the exponential growth model $A = A_0 e^{kt}$. Match each mathematical expression used in this evaluation process to its corresponding conceptual role.

An operations manager is tracking the exponential growth in demand for a new product, which increased from ${}100$ units to ${}500$ units over ${}6$ weeks. To solve for the growth rate $k$ in the resulting simplified equation ${}5 = e^{6k}$, the manager must apply the natural logarithm ($\ln$) to both sides of the equation.

Projecting Infrastructure Capacity

A cybersecurity analyst at a corporate network is modeling the continuous, exponential spread of a localized malware infection. To predict the future number of compromised devices, the analyst uses the continuous exponential growth model $A = A_0 e^{kt}$. After substituting the initial and current device counts, the equation simplifies to ${}5 = e^{6k}$. To isolate the exponent and solve for the continuous growth rate $k$, the analyst must apply the ____ to both sides of the equation.