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?

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.

Evaluating a Customer Support Ticket Growth Model