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What makes a function exponential

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Exponential functions look similar to functions we learn before, in which were involved exponents, but there is a big difference, in exponentials the variable is the power and not a base. Previously, you have dealt with such functions as f(x) = x^2, where the variable x was the base and the number 2 was the power. In the case of exponential functions, we are dealing with functions such as g(x) = 2^x, where the base is the fixed number, and the power is the variable.