Math Help - Log Sketches

1. Log Sketches

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

2. Originally Posted by xwrathbringerx

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)