| What if we really wanted to find out what that exponent was | ||
| in the last example on the last page? | ||
| Remember it? | ||
|
7log7500 = 500 |
||
| So how would we find the answer? | ||
| We could start the way we did on the log5125 problem in the last page: | ||
|
|
||
| OOPS! Three wasn't enough, and 4 was too many. | ||
| 343 is a lot closer to 500 than 2,401. | ||
| That means the answer is closer to 3 than to 4. | ||
| We know it's: | ||
|
3.(something) |
||
| But what? | ||
| We probably can't find a calculator with a base 7 log key. | ||
| We have to find a way to convert base 7 logs to base 10 logs | ||
| so we can use a calculator or we're stuck. | ||
| You're in luck! We just happen to have a formula to do that. | ||
| (Somehow I just knew we would!) | ||
| So if we have something like: | ||
|
log7500 |
||
| We want to change this into a base 10 log. | ||
| To do this, we make a fraction. | ||
| The fraction has a base 10 log on the top (numerator). | ||
| We put another base 10 log on the bottom (the denominator). | ||
| We have: | ||
|
|
||
| That's all there is to it! | ||
| Now if we have a calculator with a log key, we can solve this puppy. | ||
|
log10500 = 2.69897 log107 = 0.845098 |
||
| So ... | ||
|
|
||
| So that means ... | ||
|
73.19368 = 500 |
||
| In general, using a and b to stand for any old numbers we might have, | ||
| we can say: | ||
|
|
||
| This trick works for any base you want to change to. | ||
| You could even say: | ||
|
|
||
| if you wanted to. | ||
| But why would you want to? | ||
|
copyright 2005 Bruce Kirkpatrick |
