Negative Bases

Consider the sequence

1, -3, 9, -27, 81, -243

The terms of the sequence alternate between positive and negative numbers.

The base is the above expressions is -3.

To solve the equationwithwe can log both sides to obtain

(1) (log here means log base 10 )

This is not a general result for real numbers. It can only be used forsince we cannot take the log of a negative number (at least when keeping to real numbers).

If we try and solve it for the equationapplying (1) we obtainwhich is not defined, since the log of a negative number is not defined (keeping to real numbers). In fact, inspection of the equationreturns the solution

We can however solve the equation by taking log base (-3) of both sides. If we do this for the equationwe obtain

Now use the identityifto giveto give

This method works in this particular case, but not generally.

The equationhas no solution in real numbers with any amount of manipulation. Solutions involving complex numbers do exist however.