Questions: 2(ln(2x)-ln(y))-(ln(3)+2ln(5))

Solution Steps

To simplify the given expression, we can use the properties of logarithms. Specifically, we can use the properties:

1. $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$
2. $a \ln(b) = \ln(b^a)$
3. $\ln(a) + \ln(b) = \ln(ab)$

Using these properties, we can simplify the expression step by step.

1. Apply the property $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ to the term inside the first set of parentheses.
2. Apply the property $a \ln(b) = \ln(b^a)$ to the term inside the second set of parentheses.
3. Combine the results using the properties of logarithms.

We start with the expression: $$2(\ln(2x) - \ln(y)) - (\ln(3) + 2\ln(5))$$

Using the properties of logarithms, we can rewrite the expression: $$2(\ln(2x) - \ln(y)) = 2\ln(2x) - 2\ln(y) = \ln((2x)^2) - \ln(y^2) = \ln\left(\frac{(2x)^2}{y^2}\right)$$ And for the second part: $$-(\ln(3) + 2\ln(5)) = -\ln(3) - \ln(5^2) = -\ln(3) - \ln(25) = -\ln(75)$$

Now we combine the two parts: $$\ln\left(\frac{(2x)^2}{y^2}\right) - \ln(75) = \ln\left(\frac{(2x)^2}{75y^2}\right)$$

Final Answer