Assume that x,y and a are positive numbers. Use the properties of logarithms to write the following expression - log a x^6 y^7
a. 42log a x+y
b. 42log a x+7log a y
c. 6log a x+42log a y
d. 6log a x+7log a y
e. 6log a x+6log a y
assuming that "a" is the base
- log a x^6 y^7
= -(loga x^6 + loga y^7)
= -(6loga x + 7loga y)
= -6loga x - 7loga y
none of your choices are correct, unless there is a typo with that minus sign
To write the expression -log a x^6 y^7 using the properties of logarithms, we can use the following rules:
1. Power Rule: log a (x^m) = m * log a (x)
2. Product Rule: log a (x * y) = log a (x) + log a (y)
3. Negative Rule: -log a (x) = log a (1/x)
Let's break down the expression step by step:
1. Start with -log a (x^6 y^7)
2. Apply the Power Rule for x^6: -6 * log a (x) + log a (y^7)
3. Apply the Power Rule for y^7: -6 * log a (x) + 7 * log a (y)
4. Now, we have -6 * log a (x) + 7 * log a (y)
Therefore, the correct answer is d. 6log a x + 7log a y.