Question
Expand completely log11(square root x+y/x^2y)
Answers
GPT 3.5
Let's use logarithmic properties to simplify the expression step by step.
First, we can rewrite the expression as log11[(x + y) / (x^2y)^(1/2)].
Using the properties of logarithms, we can rewrite the square root as an exponent: log11[(x + y) / ((x^2y)^(1/2))] = log11[(x + y) / (x^(1/2) * y^(1/2))].
Next, we can simplify the expression further by dividing the terms inside the logarithm: log11[(x + y) / (x^(1/2) * y^(1/2))] = log11(x + y) - log11(x^(1/2) * y^(1/2)).
Since log11(x^(1/2) * y^(1/2)) = log11(x^(1/2)) + log11(y^(1/2)), we can simplify further: log11(x + y) - log11(x^(1/2) * y^(1/2)) = log11(x + y) - [log11(x^(1/2)) + log11(y^(1/2))].
Finally, we can simplify each term by using the logarithmic properties, leading us to the fully expanded form:
log11(x + y) - [log11(x^(1/2)) + log11(y^(1/2))] = log11(x + y) - [1/2 * log11(x) + 1/2 * log11(y)].
First, we can rewrite the expression as log11[(x + y) / (x^2y)^(1/2)].
Using the properties of logarithms, we can rewrite the square root as an exponent: log11[(x + y) / ((x^2y)^(1/2))] = log11[(x + y) / (x^(1/2) * y^(1/2))].
Next, we can simplify the expression further by dividing the terms inside the logarithm: log11[(x + y) / (x^(1/2) * y^(1/2))] = log11(x + y) - log11(x^(1/2) * y^(1/2)).
Since log11(x^(1/2) * y^(1/2)) = log11(x^(1/2)) + log11(y^(1/2)), we can simplify further: log11(x + y) - log11(x^(1/2) * y^(1/2)) = log11(x + y) - [log11(x^(1/2)) + log11(y^(1/2))].
Finally, we can simplify each term by using the logarithmic properties, leading us to the fully expanded form:
log11(x + y) - [log11(x^(1/2)) + log11(y^(1/2))] = log11(x + y) - [1/2 * log11(x) + 1/2 * log11(y)].