Problem: If logM​N=logN​M,Mî€ =N,MN>0,Mî€ =1,Nî€ =1, then MN equals:
Answer Choices:
A. 21​
B. 1
C. 2
D. 10
E. a number greater than 2 and less than 10
Solution:
Let logM​N=x; then logN​M=logM​N1​=x1​.∴x2=1,x=+1 or -1 . If x=1 then M=N, but this contradicts the given Mî€ =N. If x=−1, then N=M−1∴MN=1
or
Let logM​N=x=logN​M∴N=Mx and M=Nx∴(Mx)x=Nx=M
∴x⋅x=1∴x=1 (rejected) or x=−1∴N=M−1∴NM=1