Logarithms: log9^x + logy^8=2 and logx^9 + log8^y=8/3

[cyberspace]

For this problem I have to solve the following system for (x,y):
log 9^x + log y^8=2
log x^9 + log 8^y=8/3

I looked in the back of my book and it said {(27,64) , (3,4)}
I have absolutely no idea how to solve this or get this answer.
Do I use system of equations for this? If so, how?

Thanks for the help, I appreciate it!

 

[stapel]

I have to solve the following system for (x,y):

log 9^x + log y^8=2
log x^9 + log 8^y=8/3

I looked in the back of my book and it said {(27,64) , (3,4)}

I have no idea either, since this is not a valid solution. (Note: I am assuming that the unstated base of the logs is "10".) For instance, checking the point (x, y) = (3, 4), we get:

. . . . .$\displaystyle \log{(9^3)}\, +\, \log{(4^8)}\, =\, \log{(9^3\,4^8)}\, =\, \log{(47775744)}\, \approx \,7.679...$

...and not "2", as required.

Eliz.

 

[cyberspace]

Sorry for the confusion,

when I typed log9^x and others, I meant "9" was the base and "x" is the argument.

Again, sorry if I've confused you.

 

[Deleted member 4993]

Sorry for the confusion,

when I typed log9^x and others, I meant "9" was the base and "x" is the argument.

Again, sorry if I've confused you.

so the problem is

$\displaystyle Log_9(x) + Log_y(8) = 2$

$\displaystyle Log_x(9) + Log_8(y) = 8/3$

hint

if

$\displaystyle Log_9(x) = m$

then

$\displaystyle Log_x(9) = 1/m$

 

[Deleted member 4993]

This problem will be easier, if one assumes

$\displaystyle Log_9(x) = m$

and

$\displaystyle Log_8(y) = n$

 

[stapel]

hint: if $\displaystyle Log_9(x) = m$

then $\displaystyle Log_x(9) = 1/m$

Note: If you're not sure about the above hint, here is a proof, using the change-of-base formula:

. . . . .log[sub:1n6v8o1y]b[/sub:1n6v8o1y](a) = c

. . . . .log[sub:1n6v8o1y]a[/sub:1n6v8o1y](a) / log[sub:1n6v8o1y]a[/sub:1n6v8o1y](b) = c

. . . . .1 / log[sub:1n6v8o1y]a[/sub:1n6v8o1y](b) = c

. . . . .1 = c log[sub:1n6v8o1y]a[/sub:1n6v8o1y](b)

. . . . .1 / c = log[sub:1n6v8o1y]a[/sub:1n6v8o1y](b)

If you get stuck, please reply showing all of your steps and reasoning, including how you used what you've been given in previous replies. Thank you!

Eliz.