True or False? If true, prove it. If false, find the true...

True or False? If true, prove it. If false, find the true answer.
The populations of New York and Los Angeles are growing at 1$\%$ and 1.4$\%$ a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?

Key Concepts

Exponential Growth

Exponential growth describes the process by which a quantity increases over time at a rate proportional to its current value. This type of growth is characterized by a constant percentage increase per time period, resulting in a multiplicative effect over time. It underpins many natural and financial processes, including populations, investments, and radioactive decay.

Compound Growth Model

The compound growth model is used to represent exponential growth mathematically. It typically takes the form P(t) = P0 * (1 + r)^t, where P0 is the initial value, r is the growth rate per time period, and t is the number of time periods. This model helps predict future values and is essential in comparing scenarios with different growth rates.

Logarithms in Solving Exponential Equations

When equating two exponential growth functions with different rates, logarithms are used to solve for the time variable. By taking the logarithm of both sides of an equation, one can isolate the exponent and determine the time at which two quantities become equal. This method is crucial when dealing with equations where the variable appears in the exponent.

Equating Exponential Models

In problems that involve comparing two quantities that grow at different exponential rates, equating the two compound growth models allows one to find the point in time when they are equal. Solving these equations typically involves isolating the time variable through logarithmic manipulation. This concept is fundamental in analyzing crossover points in growth trends.

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