Homework Help: Summation of a Logarithmic Series

1. Jul 10, 2012

S.R

1. The problem statement, all variables and given/known data
What is the sum of the following series?

log(3/2)+log(4/3)+log(5/4)+...log(200/199).

Where log(x) is log base 10 of x.

2. Relevant equations

3. The attempt at a solution
Evidently, the previous form equals:

log(3/2*4/3*5/4*...200/199)

I'm missing something - no patterns are evident to me other than the denominator and numerator of the subsequent terms cancel out.

Any guidance would be appreciated.

2. Jul 10, 2012

Curious3141

And that's important! What are you left with after cancelling *everything* that can be cancelled?

3. Jul 10, 2012

Saitama

$$log(\frac{3}{2}*\frac{4}{3}*\frac{5}{4}*\frac{6}{5}..........*\frac{200}{199})$$

Do you see in what pattern the terms cancel and which terms are left?

4. Jul 10, 2012

S.R

Intuitively, yes - would it be correct in saying that all terms cancel other than 1/2 and 200/1, leaving 200/2?

5. Jul 10, 2012

Mentallic

Yes, that would be correct, and you didn't find it intuitive even though you spotted the pattern?

6. Jul 10, 2012

Saitama

Yes, only 1/2 and 200/1 remains. 3 cancels 1/3, 4 cancels 1/4, 5 cancels 1/5 but there's no one to cancel 200 and 1/2.

7. Jul 10, 2012

S.R

Quickly edited after rereading . I'm not sure why I didn't recognize that pattern before.