1951 AMC 12 Problem 35

Question

If

a^x=c^q=b

and

c^y=a^z=d,

then:

xy=qz

\dfrac{x}{y}=\dfrac{q}{z}

x+y=q+z

x-y=q-z

x^y=q^z

Take logarithms of both chains of equal powers

From

x\log a=q\log c

and

y\log c=z\log a

, eliminate the logarithms

Taking logarithms gives

\begin{gathered} x\log a=q\log c,\\ y\log c=z\log a. \end{gathered}

Multiplying these equations and cancelling the nonzero logarithmic factors under the historical nondegenerate-base convention yields

xy=qz.

Thus, the intended correct answer is A.

Problem 35 in other years