Using an identity to simplify the sum

Question

So I ran into this problem today. It asks me to use an identity to simplify the sum.

$$\sum_{j=7}^{27}\ln\left(\frac{j+1}{j}\right)$$

I have no idea where to start. I don't know any identity that fits this formation. Thanks.

Answer 1

Hint:

$$ \ln \dfrac a b = \ln a - \ln b$$

Answer 2

Hint:

$$\ln\left(\frac{j+1}{j}\right)=\ln(j+1)-\ln(j)$$

This for $j>0$.

Answer 3

You can also look at the product; since $$A=\sum_{j=7}^{27}\ln\left(\frac{j+1}{j}\right)$$ then $$e^A=\prod_{j=7}^{27}\frac{j+1}{j}=\frac{8\times 9\times 10\times 11 \cdots \times 28}{7\times 8\times 9\times 10\times 11 \cdots \times 27}=\frac{(8\times 9\times 10\times 11 \cdots\times 27) \times 28}{7\times (8\times 9\times 10\times 11 \cdots \times 27)}$$ and notice how easily it simplifies. What is left is $$e^A=\frac{28}{7}=4$$ So, $A=2\log(2)$