Understanding Exponential Functions: A Comprehensive Guide to Growth and Decay

Exponential Function

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

An exponential function is a mathematical function of the form f(x) = a^x, where a is a constant and x is the exponent. The constant “a” is called the base of the exponential function.

In an exponential function, the variable x is in the exponent, which means that as x increases, the function grows or decays exponentially. The value of the base “a” determines how fast or how slowly the function grows or decays.

If a > 1, the exponential function will exhibit exponential growth. As x increases, the function increases rapidly, becoming steeper and steeper. For example, f(x) = 2^x is an exponential growth function with a base of 2. As x increases, the function values double.

If a is a fraction between 0 and 1, the exponential function will exhibit exponential decay. As x increases, the function decreases rapidly, getting closer and closer to zero. For example, f(x) = (1/3)^x is an exponential decay function with a base of 1/3. As x increases, the function values decrease.

It is important to note that the domain of an exponential function is all real numbers, and the range depends on the base “a.” If a > 1, the range is (0, infinity), meaning that the function values are positive and approach infinity as x increases. If 0 < a < 1, the range is (0, infinity), meaning that the function values are positive and approach zero as x increases. Exponential functions have various applications in real life, such as population growth, compound interest, radioactive decay, or the spread of diseases. They are also widely used in science, engineering, and finance. To solve problems involving exponential functions, you can use logarithms to manipulate the equation. The logarithm of a number with respect to a given base "a" gives you the exponent to which the base must be raised to obtain that number. By taking logarithms of both sides of an exponential equation, you can solve for the variable.

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