# Math Help - Logarithm Problem I can't figure out

1. ## Logarithm Problem I can't figure out

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

2. Originally Posted by ConMan
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.