Solving Simultaneous Equations Using Logarithms

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Homework Help Overview

The problem involves solving simultaneous equations that include logarithmic expressions and exponential functions. The equations presented are 2log₂y = log₄3 + log₂x and 3^y = 9^x, which require manipulation of logarithmic properties and exponential relationships.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to manipulate the equations by expressing 9^x in terms of 3^y and relates y to x. Some participants suggest using properties of logarithms to simplify the expressions further, while others note the relationship between logarithmic and exponential forms.

Discussion Status

Participants have provided guidance on how to proceed with the logarithmic manipulations and have confirmed the correctness of the initial steps taken by the original poster. There is an ongoing exploration of the implications of the logarithmic identities mentioned.

Contextual Notes

There appears to be some uncertainty regarding the next steps in solving for x and y, as well as the application of logarithmic properties. The original poster expresses a lack of clarity on how to proceed from their current position.

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Homework Statement


Solve the simultanious equations
2log_2y = log_43 + log_2x
3^y = 9^x

The Attempt at a Solution


3^y = (3^2)^x
y = 2x

2log_2 x + 2log_2 2 = log_2 \sqrt{3} + log_2 x
log_2 x = log_2 \frac{\sqrt{3}}{4}


No idea what to do from where, what does x equals?
 
Last edited by a moderator:
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Correct up to now.
The only steps that remain to be done is to use

2log(...) = log[(...)^{2}] and

logu + logv = log (uv)
 
If logax = logay, then x = y.
 
Ah okay, thank you guys.
 

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