For some values of r_{i} the Moran equation can be solved exactly,
but often we must solve it numerically. We illustrate both methods for the equation |

0.5^{d} + 0.5^{d} + 0.5^{d} + 0.25^{d} = 1 |

Numerical solution Most graphing calculators and computer algebra
systems have routines for numerical solution of equations. For example, in
Mathematica the command is |

FindRoot[.5^d + .5^d + .5^d + .25^d == 1,{d,1}] |

The {d,1} specifies that d is the variable, and provides a starting guess of 1 for the solution. |

For this example, the FindRoot command gives the solution 1.72368. |

Exact solution This instance of the Moran equation can be solved
analytically when we note ^{2}. |

3⋅(1/2)^{d} + (1/4)^{d} = 1 |

can be written as |

3⋅(1/2)^{d} + ((1/2)^{2})^{d} = 1 |

Interchanging the exponents of the second term of the left side we obtain |

3⋅(1/2)^{d} + ((1/2)^{d})^{2} = 1 |

Writing |

(1/2)^{d} = x, |

the Moran equation becomes the quadratic equation |

3x + x^{2} = 1. |

The quadratic formula gives |

x = (-3 ± √(13))/2. |

Recalling ^{d} |

x = (-3 + √(13))/2 |

and so |

d = Log((-3 + √(13))/2)/Log(1/2). |

This approach can be adopted to any situation in which all the scaling factors are (integer) powers of one of the scaling factors. |

Return to the Moran equation.