# Thread: how much paper would I need to write out this number?? (applied use of logarithms)

1. ## how much paper would I need to write out this number?? (applied use of logarithms)

Got this problem to do....... don't know where to start. Need help!

If 9^(99) is multiplied out and typed on a strip of paper, 3 digits per centimeter, about how many kilometers long would the paper be?

A hint says use the common log to determine the number of digits the number has, but I don't understand what that means

2. ## Re: how much paper would I need to write out this number?? (applied use of logarithms

The common log is base 10.

3. ## Re: how much paper would I need to write out this number?? (applied use of logarithms

Yes I know that, but I forgot to mention one thing, I revised my post so I'm a bit clearer.

4. ## Re: how much paper would I need to write out this number?? (applied use of logarithms

If there are $x$ digits in $9^{9^9}$ , then solve for $x$ in $10^{x-1}=9^{9^9}$.

Remember to round your value of $x$ down, because you cannot have part of a digit.

5. ## Re: how much paper would I need to write out this number?? (applied use of logarithms

So I'd solve for x and then use a conversion factor to get the length of the paper? Also, where does the 10^(x-1) come from?

6. ## Re: how much paper would I need to write out this number?? (applied use of logarithms

Originally Posted by zachd77
So I'd solve for x and then use a conversion factor to get the length of the paper? Also, where does the 10^(x-1) come from?
$10^{x-1} = 9^{9^9} \implies x-1 = 9^9 \log_{10}(9) \approx 369693099$, (rounding down to the nearest whole number).

Add 1 to both sides, and we get

$x=369693100$.

Three digits per centimeter means you will need

$\frac{369693100}{3} = 123231033.\overline{33} \text{ centimeters.}$

Converting from cm to km, the answer is approximately 1232.31 km.

7. ## Re: how much paper would I need to write out this number?? (applied use of logarithms

Wow.. thanks again!!