Expand the Logarithmic Expression log of (5x)/(4y)

Problem

(log_)((5*x)/(4*y))

Solution

1. Apply the quotient rule for logarithms, which states that (log_)(A/B)=(log_)(A)−(log_)(B)

(log_)((5*x)/(4*y))=(log_)(5*x)−(log_)(4*y)

2. Apply the product rule for logarithms to both terms, which states that (log_)(A*B)=(log_)(A)+(log_)(B)

(log_)(5*x)=(log_)(5)+(log_)(x)

(log_)(4*y)=(log_)(4)+(log_)(y)

3. Substitute the expanded product terms back into the expression, ensuring the subtraction is distributed to both parts of the second term.

(log_)(5)+(log_)(x)−((log_)(4)+(log_)(y))

4. Distribute the negative sign to simplify the final expression.

(log_)(5)+(log_)(x)−(log_)(4)−(log_)(y)

Final Answer

(log_)((5*x)/(4*y))=(log_)(5)+(log_)(x)−(log_)(4)−(log_)(y)

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