An operations manager uses the growth model ${}2e^{x-2} = 18$ to predict the number of months ($x$) required for a new branch to reach its target efficiency. Arrange the following steps in the correct sequence to solve for $x$.

An operations analyst is using exponential equations to model system growth. Match each mathematical component or step used in solving equations such as ${}2e^{x-2} = 18$ and ${}5e^{2x} = 25$ with its correct simplified value or description.

A small business owner is using the growth model ${}2e^{x-2} = 18$ to predict when monthly revenue will reach a specific target. True or False: To correctly use the property $\ln e^u = u$ to solve for $x$, the owner must first divide both sides of the equation by 2 to isolate the exponential term.

A data analyst is solving the exponential equation $e^{2x} = 5$ to determine when a server's processing load will reach capacity. They take the natural logarithm of both sides to get $\ln e^{2x} = \ln 5$. To simplify the left side of the equation to just ${}2x$, the analyst uses the fundamental mathematical fact that $\ln e$ evaluates to ____.

Steps for Solving an Exponential Revenue Model

Logarithmic Simplification in Customer Traffic Modeling