Logs two equtions

**mathwhat** · April 24th 2010, 04:54 PM

$\log_3(x)+\log_9(y)=1$
$\log_x(3)+\log_y(9)=-\frac{1}{2}$

How can i solve this?

Thank you!

**Prove It** · April 24th 2010, 05:13 PM

Originally Posted by mathwhat

$\log_3(x)+\log_9(y)=1$
$\log_x(3)+\log_y(9)=-\frac{1}{2}$

How can i solve this?

Thank you!

Are you asking to solve these equations simultaneously for $x$ and $y$ ?

I'd try converting everything to the natural logarithm using the change of base rule $\log_b{x} = \frac{\log_k{x}}{\log_k{b}}$ .

So Equation 1:

$\log_3{x}+\log_9{y}=1$

$\frac{\ln{x}}{\ln{3}} + \frac{\ln{y}}{\ln{9}} = 1$

$\frac{\ln{x}}{\ln{3}} + \frac{\ln{y}}{2\ln{3}} = 1$

$\frac{2\ln{x} + \ln{y}}{2\ln{3}} = 1$

$2\ln{x} + \ln{y} = 2\ln{3}$

$\ln{y} = 2\ln{3} - 2\ln{x}$ .

Equation 2:

$\log_x{3}+\log_y{9}=-\frac{1}{2}$

$\frac{\ln{3}}{\ln{x}} + \frac{\ln{9}}{\ln{y}} = -\frac{1}{2}$

$\frac{\ln{3}}{\ln{x}} + \frac{2\ln{3}}{\ln{y}} = -\frac{1}{2}$

$\frac{\ln{3}\ln{y} + 2\ln{3}\ln{x}}{\ln{x}\ln{y}} = -\frac{1}{2}$

$2\ln{3}\ln{y} + 4\ln{3}\ln{x} = -\ln{x}\ln{y}$

$2\ln{3}\ln{y} - \ln{x}\ln{y} = -4\ln{3}\ln{x}$

$\ln{y}(2\ln{3} - \ln{x}) = -4\ln{3}\ln{x}$

$\ln{y} = \frac{-4\ln{3}\ln{x}}{2\ln{3} - \ln{x}}$ .

Now you can set the equations equal to each other and solve.

**mathwhat** · April 24th 2010, 08:03 PM

Thank you but i don't familiar with $\ln$ ...
I am only know the $\log$ thing...

**Prove It** · April 24th 2010, 08:06 PM

Originally Posted by mathwhat

Thank you but i don't familiar with $\ln$ ...
I am only know the $\log$ thing...

$\ln$ is just shorthand for the logarithm with base $e$ .

I.e. $\ln{x} = \log_e{x}$ .

If you want you can use the base 10... It doesn't make any difference. The point of using the change of base rule is to get all the logarithms to HAVE THE SAME BASE.

**mathwhat** · April 25th 2010, 07:52 AM

Thank you very much for the useful information!!!

**icefirez** · April 29th 2010, 06:59 AM

hi this is my answer i dont know its right cuz im not good at math...but hopefully it's right from my mention
just anohter thing i dont know to much latex language so ....

$ log_3{x}+ log_9{y} = 1 $
$ log_3{x}+log_3^2{y}=1 $
here is log base 3 SQUARED of y but cant do it with latex
$ \log_3{9}=\log_3{3^2}=2\log_3{3}=2$
its another role of changing base now example
$\log_9{3}=\log_3^2{3}=\frac{1}{2}\log_3{3}=\frac{1 }{2}$

$ log_3{x}+\frac{1}{2}log_3{y}=log_3{3} $

we know that log base 3 of number 3 is 1 so we can write 1 as log base 3 of 3

now we remove logs and we got

$ x+\frac{1}{2}y=1 $
hope im right and you solve other equation

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