log-system

May 2009
596
31
ALGERIA
Solve the system

\(\displaystyle log_{a^{2}}(x)-log_{a^{4}}(y)=3\)
\(\displaystyle log_{a^{6}}(x)+log_{a^{8}}(y)=4\)
 

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MHF Hall of Honor
Mar 2010
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Chicago
Solve the system

\(\displaystyle log_{a^{2}}(x)-log_{a^{4}}(y)=3\)
\(\displaystyle log_{a^{6}}(x)+log_{a^{8}}(y)=4\)
Is \(\displaystyle a\) a constant?

Here are some things you can do to the first equation.

\(\displaystyle log_{a^{2}}(x)-log_{a^{4}}(y)=3\)

\(\displaystyle \Longrightarrow \frac{log_a(x)}{log_a(a^2)}-\frac{log_a(y)}{log_a(a^4)}=3\)

\(\displaystyle \Longrightarrow \frac{log_a(x)}{2}-\frac{log_a(y)}{4}=3\)

\(\displaystyle \Longrightarrow 2log_a(x)-log_a(y)=12\)

\(\displaystyle \Longrightarrow log_a(x^2)-log_a(y)=12\)

\(\displaystyle \Longrightarrow log_a\left(\frac{x^2}{y}\right)=12\)

Keep in mind that \(\displaystyle x>0\) and \(\displaystyle y>0\).
 
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Soroban

MHF Hall of Honor
May 2006
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Lexington, MA (USA)
Hello, dhiab!

Edit: I've corrected my blunder . . .


Solve the system:

. . \(\displaystyle \log_{a^2}(x)-\log_{a^4}(y) \;=\; 3 \;\;[1] \)

. . \(\displaystyle \log_{a^6}(x)+\log_{a^8}(y)\:=\:4\;\;[2]\)

We have:

. . \(\displaystyle \frac{\ln x}{\ln a^2} \;{\color{red}-}\; \frac{\ln y}{\ln a^4} \;=\;3 \quad\Rightarrow\quad \frac{\ln x}{2\ln a} \;-\; \frac{\ln y}{4\ln a} \;=\;3 \quad\Rightarrow\quad 2\ln x \;-\; \ln y \;=\;12\ln a \;\;[1]\)

. . \(\displaystyle \frac{\ln x}{\ln a^6} \;+\; \frac{\ln y}{\ln a^8} \;=\;4 \quad\Rightarrow\quad \frac{\ln x}{6\ln a} \;+\; \frac{\ln y}{8\ln a} \;=\;4 \quad\Rightarrow\quad 4\ln x \;+\; 3\ln y \;=\;96\ln a \;\;[2]\)



\(\displaystyle \begin{array}{ccccccc}3\times [1]: & 6\ln x -3\ln y &=& 36\ln a \\
\text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}\)

And we have: .\(\displaystyle 10\ln x \:=\:132\ln a \quad\Rightarrow\quad \ln x \:=\:\tfrac{66}{5}\ln a \:=\:\ln\left(a^{\frac{66}{5}}\right)\)

. . Therefore: .\(\displaystyle \boxed{x \;=\;a^{\frac{66}{5}}}\)



\(\displaystyle \begin{array}{ccccc}\text{-}2\times [1]\!: & \text{-}4\ln x + 2\ln y &=& \text{-}24\ln a \\
\text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}\)

And we have: .\(\displaystyle 5\ln y \:=\:72\ln a \quad\Rightarrow\quad \ln y \:=\:\tfrac{72}{5}\ln a \:=\: \ln\left(a^{\frac{72}{5}}\right)\)

. . Therefore: .\(\displaystyle \boxed{y \;=\;a^{\frac{72}{5}}}\)

 
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May 2009
596
31
ALGERIA
Hello, dhiab!

It may be neater with natural logs . . . or maybe not.


We have:

. . \(\displaystyle \frac{\ln x}{\ln a^2} + \frac{\ln y}{\ln a^4} \;=\;3 \quad\Rightarrow\quad \frac{\ln x}{2\ln a} + \frac{\ln y}{4\ln a} \;=\;3 \quad\Rightarrow\quad 2\ln x + \ln y \;=\;12\ln a \;\;[1]\)

. . \(\displaystyle \frac{\ln x}{\ln a^6} + \frac{\ln y}{\ln a^8} \:=\:4 \quad\Rightarrow\quad \frac{\ln x}{6\ln a} + \frac{\ln y}{8\ln a} \:=\:4 \quad\Rightarrow\quad 4\ln x + 3\ln y \;=\;96\ln a \;\;[2]\)


\(\displaystyle \begin{array}{ccccccc}\text{-}3\times [1]: & \text{-}6\ln x -3\ln y &=& \text{-}36\ln a \\\)\(\displaystyle
\text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}\)

And we have: .\(\displaystyle -2\ln x \:=\:60\ln a \quad\Rightarrow\quad \ln x \:=\:-30\ln a \:=\:\ln\left(a^{-30}\right)\)

. . Therefore: .\(\displaystyle \boxed{x \;=\;a^{-30}}\)


\(\displaystyle \begin{array}{ccccc}\text{-}2\times [1]\!: & \text{-}4\ln x - 2\ln y &=& \text{-}24\ln a \\\)\(\displaystyle
\text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}\)

And we have: .\(\displaystyle \ln y \:=\:72\ln a \quad\Rightarrow\quad \ln y \:=\:\ln\left(a^{72}\right)\)

. . Therefore: .\(\displaystyle \boxed{y \;=\;a^{72}}\)
HELLO : thank you but in first equation you have - not +(Evilgrin)