linear relationship

M=log₉E means that 9^M = E

 

N = log₂₇E means that 27^N = 2E   or   27^N/2 = E

 

This means 9^M = 27^N/2

 

Take logs of both sides (I'll use log to the base e, or ln):  ln(9^M) = ln(27^N/2)

 

Using a property of logarithms this can be written as: ln(9^M) = ln(27^N) - ln(2)

 

Using another property of logarithms: M*ln(9) = N*ln(27) - ln(2) or:   M = N*ln(27)/ln(9) - ln(2)/ln(9)

 

This is in the form y = m*x + c, which is the general form of a linear equation (here y = M, x = N, m = ln(27)/ln(9),

c = -ln(2)/ln(9)), so the answer is yes.

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