Exponential function , Power function

　・In Japanese
＜Premise knowledge＞
　・Logarithmic function

■What is a exponential?

The exponent is defined as follows.

■What is a power function?

The power function is a kind of exponential function and is expressed by the following formula.The base is a variable and the exponent is a constant.



When k = -1, it is inversely proportional, and when K = 1, it is directly proportional.

＜Power law＞
Power law is one of the statistical models, and the characteristics of k <0 and x> 0 in the above equation can be expressed by a probability distribution. The part that decays is called the long tail.

■What is an exponential function?

The exponential function is expressed by the following formula. It shows the comparison with the power function (red line is the exponential function). The exponential function converges or diverges faster depending on the value of x.



The function when k = e (number of Napiers) is frequently used.



The following is called a sigmoid function and is used in neural networks. There are also tanh functions.

■Expressing power functions and exponential functions in logarithms

When the power function expresses the x-axis and y-axis with the logarithmic axis, and the exponential function expresses only the y-axis with the logarithmic axis, it shows linear characteristics as shown below. By using this property, it is possible to judge whether data with a certain characteristic follows a power function or exponential function by seeing whether it is linear in a logarithmic graph.



The reason for being linear is as follows. You can see by expressing each function logarithmically.

＜scale invariance of power law functions＞

The following is "a" comparison of the characteristics of the power function when "a" is changed. This means that the degree of attenuation is the same even if the value of "a" is different. This is called scale invariance.

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