How to proceed: Log Equation System

Markklein

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Jul 2, 2016
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Good morning,

Please, I'm currently learning logarithms after a long time. I got stuck with an apparently simple problem that I have no idea how to solve, namely:

"Being lg2 and lg4 the logarithmics functions of base 2 and 4, respectively. It's known that lg2(x) + lg4(y) = 2 and lg4x + lg2y = 1. Calculate x + y"

I've tried to solve it by adding the two equations (changed both x and y bases), but I had this weird feelings that I'm messing up everything and will end up with an even weirder solution.

How should I proceed in such a case?

ANY help will be highly appreciated!

Thank you so much!
 
"Being lg2 and lg4 the logarithmics functions of base 2 and 4, respectively. It's known that lg2(x) + lg4(y) = 2 and lg4x + lg2y = 1. Calculate x + y"

I've tried to solve it by adding the two equations (changed both x and y bases), but I had this weird feelings that I'm messing up everything and will end up with an even weirder solution.
Please reply showing your efforts so far, starting with your change of base, such as:

. . . . .\(\displaystyle \log_2(x)\, +\, \log_4(y) \, =\, \log_2(x)\, +\, \dfrac{\log_2(y)}{\log_2(4)}\, =\, \log_2(x)\, +\, \dfrac{1}{2}\, \log_2(y)\, =\, \log_2(4)\)

You did something similar to the other given equation. Then you applied log rules (here) to consolidate the left-hand sides, equated the arguments, and... then what?

Please be complete. Thank you! ;)
 
Last edited:
. . . . .\(\displaystyle \log_2(x)\, +\, \log_4(y) \, =\, \log_2(x)\, +\, \dfrac{\log_2(x)}{\log_2(4)}\, = \ ...\ \)

.\(\displaystyle \log_2(x)\, +\, \log_4(y) \, =\, \log_2(x)\, +\, \dfrac{\log_2(y)}{\log_2(4)}\, \ =\ ...\)
 
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