Logarithm Problem I can't figure out

ConMan · #1 $Old$ November 3rd, 2009, 07:08 PM

These questions are from several math contests, and I'm trying to do these but I can't figure it out.

1. If x and y > 0 , log(base y)x + log(base x)y = 10/3 ,and (x)(y)=144
,find x + y /2

2. How many real numbers x satisfy the equation (1/5) log(base 2)x = sin(5pi x) ?

I tried making y=144/x and plugging that into the equation, but I have no idea what to do for the second question

Any help from you guys is greatly appreciated

earboth · #2 $Old$ November 4th, 2009, 02:26 AM

Quote:

Originally Posted by ConMan $View Post$

These questions are from several math contests, and I'm trying to do these but I can't figure it out.

1. If x and y > 0 , log(base y)x + log(base x)y = 10/3 ,and (x)(y)=144
,find x + y /2

...

1. You are supposed to know that $\log_b(a)=\dfrac1{\log_a(b)}$

2. Using this rule your equation becomes:

$\log_y(x)+\dfrac1{\log_y(x)} = \dfrac{10}3$

Use the substitution $z = \log_y(x)$ . Then you have:

$z-\dfrac1z = \dfrac{10}3~\implies~z^2-\dfrac{10}3 z + 1 = 0$

which yields $z = 3~\vee~z = \dfrac13$

3. Re-substituting yields:

$x = y^3~\vee~x = \sqrt[3]{y}$

4. Now use the second equation $x\cdot y = 144$ to calculate the values of x and y.

ConMan · #3 $Old$ November 4th, 2009, 04:08 PM

Thank you for your timely reply earboth, I really appreciate your help

Also, does anyone know about some online resources regarding some properties/rules on logarithms?

EDIT: this is nice: http://en.wikipedia.org/wiki/List_of...mic_identities

Defunkt · #4 $Old$ November 4th, 2009, 04:26 PM

Logarithm - Wikipedia, the free encyclopedia

Wikipedia, man's best friend!

ConMan · #5 $Old$ November 5th, 2009, 03:15 PM

Okay scratch the second question, since sin of (pi) is equal to zero. (Doh!)