Log Sketches

xwrathbringerx · #1 $Old$ 11-19-2008, 01:05 AM

Given the pts A (a, log a) and B (b, log b) on the curve y= log x, a is not equal to b.

(c) Given point Q is directly to the left of M on y = log x

(i) Find the coordinates of Q.

(ii) Show that x^2 with the base of Q = ab

For (i), I was wondering: For the y coordinate, I found it to be log ((a+b)/2). I was wondering how the answer becomes 1/2log ab (that’s what my teacher got). If it’s wrong, could someone please show me how to simplify it?

For (ii), I have no clue how to solve the question. Could someone please guide me?

N.B. For all the logs in this question, they are in base 10. Sorry

earboth · #2 $Old$ 11-19-2008, 09:32 AM

Quote:

Originally Posted by xwrathbringerx $View Post$

Given the pts A (a, log a) and B (b, log b) on the curve y= log x, a is not equal to b.

(c) Given point Q is directly to the left of M on y = log x

(i) Find the coordinates of Q.

(ii) Show that x^2 with the base of Q = ab

For (i), I was wondering: For the y coordinate, I found it to be log ((a+b)/2). I was wondering how the answer becomes 1/2log ab (that’s what my teacher got). If it’s wrong, could someone please show me how to simplify it?

For (ii), I have no clue how to solve the question. Could someone please guide me?

N.B. For all the logs in this question, they are in base 10. Sorry

M is the midpoint of A and B. Use the midpoint formula:

$M\left(\dfrac{a+b}2\ ,\ \dfrac{\log a + \log b}{2} \right)$

yields:

$M\left(\dfrac{a+b}2\ ,\ \dfrac{\log(a\cdot b)}{2} \right)$

$M\left(\dfrac{a+b}2\ ,\ \log(\sqrt{a\cdot b)} \right)$

Maybe you can use the last line to solve (ii)