## 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 equation with we 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 for since we cannot take the log of a negative number (at least when keeping to real numbers).

If we try and solve it for the equation applying (1) we obtain which is not defined, since the log of a negative number is not defined (keeping to real numbers). In fact, inspection of the equation returns the solution We can however solve the equation by taking log base (-3) of both sides. If we do this for the equation we obtain  Now use the identity if to give to give This method works in this particular case, but not generally.

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