Solve a in a power n equal b
Finding the value of a in equation (1):
an = b | (1) |
can be resolved using logarithms.
First, transform equation (1) to equation (2):
ln(an) = 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:
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(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:
a12 = 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:
1.059512 = 2.000836183
Note that equation (3) is also useful to determine n if you have a and b:
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(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:
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(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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