Answer:
the second option and x=20
Firts, Let

be the repeated decimal that we are trying to convert , so

equation (1)
Next, lets find how many digits are repeating:
It is pretty cleat that 90 is repeating, and 90 has two digits. So we are going to multiply our equation by 100 to move the decimal point two places:

(2)
Subtract equation (1) from equation (2):


Solve for


We can conclude that 10.9090909091... expressed as a rational number, i<span>n the form pq where p and q are positive integers with no common factors, is </span>

.
The logarithm written as a sum of logarithm and simplified as much as possible is 
<h3>Simplifying Logarithms</h3>
From the question, we are to write the given logarithm expression as a sum or difference of logarithms
The given logarithm is

This can be written as

From one of the rules of logarithm, we have that

Thus,
becomes

This can be further simplified into


If desired, this can be further simplified into




Hence, the logarithm written as a sum of logarithm and simplified as much as possible is 
Learn more on Simplifying logarithms here: brainly.com/question/17851187
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