# To the power of

1. Sep 8, 2009

### alfie254

Ok, I have a number to the power of another, say 2^3 which of course is 8.
Question is if you have 2^n = 8 how do you calculate n?, this is obviously simple for the example but say if I needed to calculate n in 2^n = 1000 how do I do this either on paper or using a calculator, I've searched the web and come up with nothing so it is either extremely complicated or extremely easy and staring me in the face but I just cant see it at 2 o'clock in the morning!

Thanks for any help
Steve

2. Sep 8, 2009

### slider142

The inverse operation of exponentiation is a logarithm. In particular, $\log_b(a) = c$ if $b^c = a$. The logarithm retrieves the exponent necessary to take the base b to the number a. Some calculators will only have the natural logarithm available (whose base b is Euler's number e), or only the logarithm in base 10. In that case, you will need the change of base formula, which you can derive from the defining property of the logarithm above, or you can discover on this page.
The logarithm is transcendental, it cannot be written as a finite combination of other elementary functions. However, you can use Newton's method and other series methods to calculate the logarithm on paper. This is tedious and unnecessary, however, as it was common to use logarithmic tables to look up logarithms, and today calculators produce more digits in a millisecond than you can calculate by hand in a few seconds.

Last edited: Sep 8, 2009
3. Sep 9, 2009

### Tac-Tics

Logs are amazing things. Here's just a few reasons:

They are the inverses of exponential functions. They have some very nice algebraic properties (log ab = log a + log b, and log a^b = b log a).

Log base 2 tells you the number of bits required to represent an integer.

Log base 10 tells you how many digits are in an integer.

Log base e (the natural logarithm or "ln") has a very clean definition in terms of calculus.

Logarithms are the basis for slide rules, the mechanical calculators used by engineers for hundreds of years prior to the invention of the pocket calculator.

4. Sep 9, 2009

### arildno

Most likely, your calculator will have a "Log" button and a "Ln" button

You can use either to solve your problem:

n=Log(1000)/Log(2), n=Ln(1000)/Ln(2)

5. Sep 9, 2009

### alfie254

Thank you everybody, now I understand.

Alfie