Overview: What Are Exponential Functions?
Exponential functions are a special type of mathematical functions that follow the formula y = ax. The constant a and the exponent x are both real numbers. In addition, a is not equal to 1 and a >0.
How Are Exponential Functions Graphed?
When exponential functions are graphed in the coordinate plane, they follow a smooth curve with some special properties. Very small changes in the exponent x cause the graph to become more and more steep. In addition, as the exponent x becomes smaller, the curve will get smaller and closer to the x axis, but it never touches it, no matter how small x becomes.
What Is the Logarithmic Function?
The logarithmic function is the inverse of the exponential function. If x =ay, then y = logax. For example, 23 = 8, and log28 = 3, whenever a is not equal to 1 and >0. Common logarithmic functions are logarithms of 10, which are often just written without the 10 in the subscript. They mean the same thing, though, so that log10x means the same thing as log x.
Why Are Exponential Functions Important?
Exponential functions are useful in many common applications, such as compounding interest. Population growth also follows an exponential function, as do many relationships in biology and economics. Radioactive decay also follows an exponential relationship.
Why Are Logarithmic Functions Important?
Logarithmic functions are a type of shorthand underlying important mathematical relationships. Many different scales, such as decibels measuring sound, are logarithmic. The pH measurement of acidity is also a logarithmic scale. They also have many useful applications in the measurement of musical intervals and perception.
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