Understanding the Periodicity of the Logarithmic Sine Function

In mathematical terms, a function is considered periodic if it repeats its values in regularly spaced intervals or periods. For the function fx logsin x, the challenge lies in determining the interval over which it repeats. Let's break down this function to understand its periodicity.

Periodicity of sin x

The sine function, sin x, is periodic with a period of 2π. This means that sin x sin(x 2π) for all values of x. However, the logarithmic function logy is defined only for positive values of y, which implies that the expression logsin x is valid only when sin x 0.

Intervals of Positive Sine

The sine function is positive in intervals such as (0, π), (2π, 3π), (4π, 5π), and so on. These intervals correspond to the first and second quadrants of the unit circle where the sine values are positive.

Period of logsin x

Given that sin x has a period of 2π, and logsin x repeats its behavior in every interval where sin x 0, the function logsin x is also periodic with a period of 2π. This periodicity is due to the fact that the sine function resets its behavior every 2π units, causing the logarithmic function to also repeat its pattern.

Therefore, the period of the function fx logsin x is 2π.

Graphical Representation

The graph of the function fx logsin x clearly illustrates its periodic behavior. The graph repeats itself every 2π units, confirming the periodicity of the logarithmic sine function. Desmos provides a visual verification of this periodicity.

Exploring Other Functions

It's important to note that the function fx sin(log x) behaves differently. Here, the sine function is applied to the logarithm of x. The fundamental period of the sine function is 2π, but due to the logarithmic transformation, the function does not satisfy the condition for periodicity. This means that sin(log x 2π) ≠ sin(log x). Therefore, the function fx sin(log x) is non-periodic.

Conclusion

The key takeaway is that while logsin x has a period of 2π, the function sin(log x) does not have a defined period and is therefore non-periodic. Understanding these differences is crucial for advanced mathematical analysis and applications.