## algebra2sucks 2 years ago Show why the log100 + log10000 = 2log(3)27

1. satellite73

compute all the numbers and see that they are equal

2. satellite73

you need \(\log(100)\) and also \(\log(10000)\) first

3. satellite73

if you have a power of ten , the log counts the zeros

4. algebra2sucks

lol i have no idea how to do this

5. ZeHanz

When log 100 = a, this means: \[10^a=100\]Now, because 100 is a power of 10, you can see very quickly what log 100 is... Same for log 10000 >> what power of 10 is it?

6. ZeHanz

If you have problems understanding logarithms, always remember one thing: LOGARITHMS are EXPONENTS

7. ZeHanz

In your problem, I also see a logarithm with base 3. I will call it b: \[\log_{3} 27=b\]So b is that logarithm, base 3 of 27. This means, because logarithms are exponents, \[3^b=27\]Now, 27, doesn't it sound familiar? Yes! It is a power of 3! In fact, it is 3³:\[3^b=3^3\]so b = 3. The right hand side of the problem doubles that number, so we have 6 there. Can you see now, why the left hand side is 6 as well?

8. algebra2sucks

Yea i guess so how do we answer this

9. ZeHanz

Begin with the left hand side, replace log(100) and log(10000) with the numbers we've got. These will add up to 6. Right hand side is 2*3=6 also. Now you have shown that they are equal...

10. algebra2sucks

11. ZeHanz

If you re-read the question, you'll see that "6" ids not the answer. You have to show that the left hand side and right hand side are equal. Although they seem very different, both sides give you 6 if you calculate them. Therefore they are equal. That is what you had to prove.

12. algebra2sucks

oh so how to i prove that with the equation right? i have no idea how to do this at all i know one side equals 6

13. algebra2sucks

how do we get 6 for the left hand side?

14. satellite73

\(\log(100)\) measn \(\log_{10}(100)\) and since \(100=10^2\) we know \(\log(100)=2\)

15. satellite73

last job is to solve \(\log(10000)\) i.e. solve \(10^y=10000\) for \(y\)

16. algebra2sucks

3 would be the y?

17. algebra2sucks

@satellite73 ?

18. ZeHanz

What makes you think it is 3?

19. algebra2sucks

because it has to equal to six idk i'm confused