1. The problem statement, all variables and given/known data Evaluate without a calculator: (log_{3}4 + log_{2}9)^{2} - (log_{3}4 - log_{2}9)^{2} 2. Relevant equations 3. The attempt at a solution (log_{3}4 + log_{2}9)^{2} - (log_{3}4 - log_{2}9)^{2} (2log_{3}2 + 2log_{2}3)^{2} - (2log_{3}2 - 2log_{2}3)^{2} And now I'm stuck....
You went from a minus to a divide, I think you confused it with log(a/b)= loga-logb Try using this fact a^{2}-b^{2} = (a+b)(a-b).
It also looks like you went from (log_{3}4 + log_{2}9)^{2} to (log_{3}4^{2} + log_{2}9^{2}), which is not true.
Hmmm. I think it's simpler than that, guys - Setting A = log_{3}4 and B = log_{2}9 (A+B)^{2} - (A-B)^{2} = (A^{2} + 2AB + B^{2}) - (A^{2} - 2AB + B^{2}) ... and later on using that log_{n}x^{k} = k.log_{n}x and that log_{a}b [itex]\times[/itex] log_{b}c = log_{a}c
Erm...Nevermind. I'm still stuck. Can someone walk me through it please? Edit: I totally forgot about factoring. I'll try that, thanks!!
try this (A+B)^2 - (A-B)^2 = X^2 - Y^2 = ( X + Y ) ( X - Y ) where X=(A+B) and Y=(A-B) and you get (A+B)^2 - (A-B)^2 =(2A) (2B) = 4AB
Okay one of my friends just told me to try changing the base. I did that, and now I have a giant mess on my hands. I have (ln2*ln4 + ln3*ln9 / ln3*ln2)^2 - (ln2*ln4 - ln3*ln9 / ln3*ln2)^2 and somehow I'm supposed to get 16 from all of that. I'm not really sure how...
No, no, NO! Don't change the base at the beginning. Do what Joffan suggested. You can change the base much later if you really want to.
Well I tried applying this: Setting A = log34 and B = log29 (A+B)2 - (A-B)2 = (A2 + 2AB + B2) - (A2 - 2AB + B2) The A2's and B2's will cancel out leaving me with 4AB which would be 4(log_{3}4 * log_{2}9) The answer in the book says I'm supposed to get 16. I tried changing the base to get 4(ln4*ln9 / ln3*ln2) I think I could maybe do 4(2ln2*2ln3 / ln3*ln2) but I'm not sure if that's correct. I think then maybe the ln2's and ln3's would cancel out leaving me with 4(2*2) which would be 16. But I'm not sure if that's correct.
Also, if you didn't want to change base, [tex] \begin{align} log_34 \times log_29 & = log_32^2 \times log_23^2\\ &= 2log_32 \times 2log_23\\ &= 4(log_32.log_23)\\ &= 4(log_33) \\ &=4 \end{align} [/tex]