To determine the time period of the given signal, we need to analyze its frequency content and find the fundamental frequency.

Given signal: x(t) = log(cos(a*pi*t))

Frequency of the signal: a = 50 Hz

To find the time period, we need to determine the fundamental frequency. The fundamental frequency is the lowest frequency component of a periodic signal.

In this case, the signal x(t) is a logarithmic function of the cosine function, which means it will have multiple frequency components. The fundamental frequency can be obtained by analyzing the cosine function within the logarithm.

Let's analyze the cosine function first:

cos(a*pi*t)

The standard form of a cosine function is cos(2πft), where f is the frequency in Hz. In this case, we have cos(a*pi*t), so we can equate it to the standard form:

a*pi*t = 2πft

From the equation, we can see that the fundamental frequency f is given by:

f = a/(2pi)

Given a = 50 Hz, substituting the value into the equation:

f = 50/(2*pi)

Simplifying further:

f = 25/pi

The time period (T) of a signal is the reciprocal of its frequency (T = 1/f). So, the time period of the signal x(t) is:

T = 1/(25/pi)

Simplifying further:

T = pi/25

Approximating the value of pi as 3.14:

T ≈ 3.14/25 ≈ 0.1256 s

Comparing this value with the given options, we can see that the closest option is 0.04 s, which is the correct answer.

Therefore, the time period of the given signal x(t) = log(cos(a*pi*t)) for a = 50 Hz is approximately 0.04 seconds.