Thread: Solve an equation system with logarithms

1. Solve an equation system with logarithms

x^log(y)+y^log(x^(1/2))=110
xy=1000

This is my attempt:
log(y)*log (x)+log (x^(1/2)*log(y)=log 110
y=1000/x

log(1000/x)*log (x)+log (x^(1/2)*log(1000/x)=log 110
log (1000)-log (x)*log (x)+log (x^(1/2)*log(1000)-log(x)=log 110

log(x)=t
log (1000)=3

(3-t)t+(t/2)(3-t)=log 110
3t^-3t^2+(3t-t^2)/2=log 110

I believe this is an incorrect solution, can you please tell where my error is? If my attempt is incorrect from the beginning, please show me how to solve it. Thank you in advance!

2. Re: Solve an equation system with logarithms

hi anna

$x^{\log(y)}+y^{\log(x^{\frac{1}{2}})}=110$

$y^{\log(x)}+y^{\log(x^{\frac{1}{2}})}=110$

$(y^{\log(x^{\frac{1}{2}})})^2+y^{\log(x^{\frac{1}{ 2}})}=110$

$\text{Let: } k=y^{\log(x^{\frac{1}{2}})}$

$\text{You get: }k^2+k=110$

can you continue it now?

3. Re: Solve an equation system with logarithms

k=-11, 10

10=y^log (x^(1/2))
log 10=(1/2) log x*log y
2=lg x*2 log y
2=log x*log y^2

Is this correct? If not, please show me the correct solution. Thank you!