Understanding Exponential Functions | Properties, Graphs, and Applications

Exponential function

An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is the variable

An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is the variable. In other words, the independent variable (x) appears as an exponent.

The base “a” is typically greater than 1, which causes the function to grow rapidly as x increases. However, the function can also be defined for 0 < a < 1, in which case the function decays rapidly as x increases. The graph of an exponential function can take different shapes depending on the value of a. When a > 1, the graph is increasing and has a steep slope. As x approaches negative infinity, the function approaches 0, and as x approaches positive infinity, the function grows without bound. On the other hand, when 0 < a < 1, the graph is decreasing and approaches 0 as x approaches positive infinity. Exponential functions have many applications in various fields, such as population growth, compound interest, radioactive decay, and bacterial growth, among others. They are also commonly used to model patterns of growth or decay that involve a constant growth rate or decay rate. To solve problems involving exponential functions, we can use logarithms to find missing values or evaluate equations. Logarithms are the inverse function of an exponential function, meaning they "undo" the exponential operation. This allows us to manipulate equations involving exponential functions and find the values of variables.

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