If log_MN=log_NM, M≠ N, MN>0, and M≠1, N≠1, then MN equals:

(E) a number greater than 2 and less than 10

1966 AMC 12 Problem 24

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24.

\log_MN=\log_NM,

M\ne N,

MN\gt0,

and

M\ne1,

N\ne1,

then

MN

equals:

\dfrac12

1

2

10

a number greater than

2

and less than

10

Write both logarithms using natural logs

The equality gives

(\ln M)^2=(\ln N)^2

; use

M\ne N

Change of base gives

\frac{\ln N}{\ln M}=\frac{\ln M}{\ln N},

(\ln M)^2=(\ln N)^2.

Equality of the logs themselves would give

M=N,

which is excluded. Hence

\ln M=-\ln N,

and

\ln(MN)=0.

Thus

MN=1.

Therefore, the correct answer is B.