Write number as a power of 10

Question

Just to clarify, I'm not interested in Standard Form/Scientific notation. Is it possible to write a number as a power of ten, so that for example it would look like this?

34677534678753577 = 10^36.864

Something like that, I'm aware that the example isn't correct, I'm just using it to show how it should look.

If it is possible, how do I calculate the power?

I have spent much time researching this, but I only ever find stuff on scientific notation

Thanks

Answer 1

Use what's called a logarithm with a base of 10 (as you want it to be a power of 10). For your given example:

$\log(34677534678753577)=16.54$

So

$34677534678753577=10^{16.54}$

or, more accurately,

$34677534678753577=10^{\log(34677534678753577)}$

Answer 2

Yes, this is what logarithms are. The logarithm (to base $10$) of a (positive) number $a$ is defined as the $x$ such that $10^x = a$. On writes $x = \log_{10}(a)$. Many calculators and programming languages will have this function. Note though that the default base might not be $10$.

Note that logarithms exists only for positive numbers (except you want to use complex numbers in the technical sense, that is you have a square-root of $-1$ etc).

For further details you can see the Wikipedia page on logarithm, or many other source on the subject.

Answer 3

If you want to write a number as a power of ten it's easy to use just the definition of logarithm: take a number N that you want to write as a power of ten; then for definition you have N=10^log(N) where the base of the logarithm is obviously 10. Notice that N needn't to be a natural, but can be any of the positive reals. You can even use another base and write N as a power of any positive real number (except for 1). Maybe you would like a more algorithmic answer but it's just a computational problem to find the actual value of log(N).

Answer 4

You want to have some number in the form $b^x$ for some exact values of $b$ and $x$. A logarithm for a base $b$ is defined such that it is an inverse for $b^x$, i.e. if you have some $x$, and take $b$ to the $x$-th power, you can take the logarithm for base $b$ of the resulting number to have your $x$ back (at least for positive $b$ and real $x$ under the normal definitions subject to useful conventions).

You have some number $34677534678753577=b^x$ and you set $b=10$.

That yields an equation in one variable: $$10^x=34677534678753577,$$ from which you can find $x$.

See what you have: you have got some number $x$, that the $10$ on the left is taken to the power of. So to have $x$ back, you need to reverse this process, i.e. take the logarithm for base $10$ of $10^x$, which is (you know) is equal to $34677534678753577$, that is:

$$x=\log_{10}(34677534678753577).$$

Loading this exact result to Wolfram|Alpha will give you the approximate numerical value, if you desire: $$\log_{10}(34677534678753577)\approx16.5400482147133085741294552031403205644432748544.$$

Please notice that this decimal representation never ends, and is never infinitely accurate.

You can also use a scientific calculator, like this one, to calculate an approximate value. If you cannot use yours to select the desired base for the logarithm, you can always first calculate the natural (base $e\approx2.718281828459$, usually goes as a $\log$ without a subscript or as $\ln$) logarithm, and then divide the resulting number by $\log(10)$ to have your $\log_{10}$ (the logarithm for base $10$).