For b = 1 the real exponential function is a constant and the derivative is zero because logeb=0,{\displaystyle \log _{e}b=0,} for positive a and b > 1 the real exponential functions are monotonically increasing (as depicted for b = e and b = 2), because the derivative is greater than zero for all arguments, and for b < 1 they are monotonically decreasing (as depicted for b = 1/2), because the derivative is less than zero for all arguments.

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