Assume that x, y and a are positive numbers. Use the properties of logarithms to write the expression loga x ^6y ^7 in terms of the logarithms of x and y.
a. 42loga^x+y
b. 42loga^x+7loga^y
c. 6loga^ x + 42loga^ y
d. 6loga^x+7loga^y
e. 6loga^x+6loga^y
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To write the expression loga x^6y^7 in terms of the logarithms of x and y, we can use the properties of logarithms. Specifically, the power rule states that loga (b^c) = c * loga (b).
Using this property, we can rewrite the expression as:
loga x^6y^7 = loga x^6 + loga y^7
Now, let's apply the power rule to each part separately:
loga x^6 = 6 * loga x
loga y^7 = 7 * loga y
Substituting back into the original expression, we get:
loga x^6y^7 = 6 * loga x + 7 * loga y
Therefore, the expression loga x^6y^7 in terms of the logarithms of x and y is 6 * loga x + 7 * loga y.
Hence, the correct answer is d. 6loga^x + 7loga^y.