Exponential Growth and Decay

Understanding Exponential Growth and Decay

• Exponential growth and decay is a critical concept in mathematics. Situations exhibiting exponential growth increase by the same percentage over equal time steps. Conversely, contexts that exhibit exponential decay decrease by the same percentage over equivalent time steps.
• The key attribute with exponential growth or decay is that the rate of change is proportional to the current amount.
• Mathematically, exponential growth or decay can be modeled with the equation y = a.b^x, where ‘a’ is the initial quantity, ‘b’ is the growth or decay factor, and ‘x’ represents time.

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Analysing Exponential Growth

• For exponential growth (b > 1):
• The initial quantity is ‘a’ when x = 0.
• The quantity increases multiplicatively by a constant factor b for each unit increase in x.

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Analysing Exponential Decay

• For exponential decay (0 < b < 1):
• The initial quantity is ‘a’ when x = 0.
• The quantity decreases multiplicatively by a constant factor b for each unit increase in x.

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Solving Exponential Growth and Decay Problems

• To solve exponential growth problems like b^x = a, rewrite them in logarithm form: x = log_b(a).
• To solve exponential decay problems like b^x = a, also, rewrite them in logarithm form: x = log_b(a).
• Watch out for the validity of your answers, certain mathematical expressions do not exist for certain values of x.

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Applications of Exponential Growth and Decay

• Exponential growth is commonly found when analysing situations of continuous growth, such as population studies or financial investments. For instance, budding yeasts, rabbit populations, or compounded interest on loans or investments.
• Exponential decay is frequently encountered when looking at diminishing processes, such as radioactive decay, depreciation of assets, or cooling/heating laws in physics.
• These mathematical models offer robust and accurate depictions of various real-world phenomena.