Question

Suppose logbx = logcx, where b ≠ c. Then the value of x can be

A) 0 only
B) 1 only
C) b^c or c^b only
D) any positive real number

Answer 1

Not A: We can't have
`x=0` because
`\log0` is undefined for a logarithm of any base.

B is true:
`\log1=0` for any base.

Not C: If
`x=b^c`, then
`\log_bb^c=c`, but
`\log_cb^c=c\log_bc` which only reduces to
`c` if
`\log_bc=1`. This can only happen if
`b=c`, however, but we've assumed otherwise.

Not D: The reasoning for C not being correct is enough to rule out this possibility.