Find the constant \(k\) so that \(\log_{y^5}(x^3) = k \cdot\log_y(x)\) for all positive real numbers \(x\) and \(y\) with \(y \neq 1.\)
log y^5 (x^3) = k log y (x)
Using the change-of base theorem
log x^3 log x^k
______ = ________
log y^5 log y
3 log x k log x
______ = _______
5 log y log y
3 log x log y = 5k log x log y
3 = 5k
k = 3/5
