Final answer:
The value of log(a⁴b²)/(c⁷) given loga=-4, logb=-12, and logc=1 by using the properties of logarithms is -47.
Step-by-step explanation:
The student asks to evaluate the expression log(a⁴b²)/(c⁷) given that loga=-4, logb=-12, and logc=1. Using the properties of logarithms that allow for the manipulation of logarithmic expressions, we break down the original expression into separate logarithms corresponding to each variable and exponent.
Using the property that allows us to bring the exponents down as coefficients (log(x^n) = n*log(x)), we restate the problem as 4*log(a) + 2*log(b) - 7*log(c). Substituting the values, we get:
4*(-4) + 2*(-12) - 7*(1) = -16 - 24 - 7 = -47.
Therefore, the value of the expression log(a⁴b²)/(c⁷) given the logarithms of a, b, and c is -47.