8. (log_(5)4+log_(4)5+2)(log_(5)4-log_(20)4)log_(4)5-log_(5)4

To solve the given expression, we need to simplify each logarithmic term and then perform the arithmetic operations.Let's start with the first term: Using the change of base formula, we can rewrite as .So, .Now, we can simplify the expression inside the parentheses: To combine the terms, we can multiply the numerator and denominator of the second term by : Simplifying further, we get: Now, let's move on to the second term: Using the change of base formula, we can rewrite as .So, .Now, we can simplify the expression inside the parentheses: To combine the terms, we can multiply the numerator and denominator of the second term by : Simplifying further, we get: Now, let's substitute the simplified expressions back into the original expression: Expanding the expression, we get: Simplifying further, we get: Combining the terms, we get: Simplifying further, we get: Combining the terms, we get: Simplifying further, we get: Combining the terms, we get: Simplifying further, we get: Combining the terms, we get: Simplifying further, we get:$(\log_{