# Solve a in a power n equal b

Finding the value of **a** in equation (1):

a^{n} = b |
(1) |

can be resolved using logarithms.

First, transform equation (1) to equation (2):

ln(a^{n}) = ln(b) |
(2) |

Now we can easily extract the power (n) by changing it into a multiplication:

n × ln(a) = ln(b) | (3) |

From equation (3), we know what the logarithm of a is equal to:

(4) |

Finally, we can write an equation as a = *something*.

(5) |

Now we can calculate **a** in any equation such as the following which could be used to know how much percent you need to multiply your money by 2 in 12 months:

a^{12} = 2 |
(6) |

The answer is a ≈ 1.059463094. If you can find a system where you make around 5.95% per month, cumulatively, then at the end of the year, you will have multiplied your money by 2 since:

1.0595^{12} = 2.000836183

Note that equation (3) is also useful to determine **n** if you have **a** and **b**:

(7) |

Equation (5) is why you see so many equations with e power something. We're calculating an **a** of some sort in those equations. This is also why I used the **ln()** function, which is called the Neperian logarithms. The letter e is the based used by the Neperian logarithms. The log() function uses base 10 by default. You can also specify the base by adding a subscript as in equation (8) which is used to determine the number of digits in a binary number:

(8) |

Notice how this is equivalent to using any base logarithm (Neperian in my example) and dividing by that same base logarithm of the base.

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