Logarithmic Properties in Proving Equations

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Homework Help Overview

The problem involves proving a logarithmic equation given the relationship between two variables, a and b, defined by the equation a^2 + b^2 = 47ab. The focus is on utilizing properties of logarithms to establish the equivalence log((a+b)/7) = 1/2(loga + logb).

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the use of logarithmic properties and question how to manipulate the initial equation to facilitate the proof. There is discussion about transforming a^2 + b^2 into a more manageable form, such as (a+b)^2, and the implications of such transformations.

Discussion Status

Several participants have shared their reasoning and steps taken to arrive at the logarithmic relationship. There is a sense of progress as some participants express understanding of the transformations involved, while others reflect on their previous challenges with similar problems.

Contextual Notes

One participant mentions having struggled with similar problems in the past, indicating a personal learning context. The discussion hints at the necessity of factoring and manipulating equations to simplify the proof process.

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Homework Statement


Given a^2+b^2=47ab provehat log((a+b)/7)=1/2(loga+logb)


The Attempt at a Solution


I am assuming it is using multiple log properties... idk how though :|
 
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What can be added to both sides such that a^2+b^2 can be become (a+b)2?
 
ooooooh so is this right?

(a+b)^2=49ab
a+b=7(ab)^(1/2)
(a+b)/7=(ab)^1/2

log((a+b)/7)=1/2(loga+logb)
?
 
(a+b)^2=49ab
a+b=7(ab)^(1/2)
(a+b)/7=(ab)^1/2

so log [ (a+b)/7 ] = log [ (ab)^(1/2)]

log (ab)^1/2 = 1/2 log ab = 1/2 (log a + log b)
 
yeah i got it :D

man that was so easy. i feel so stupid took 0/6 on my test ha :|

thanks for the help astronuc
 
Yeah - one just has to learn to think about factoring things and sometimes adding or subtracting from both sides of an equation to simplify it, or add, subtract and regroup on the same side.
 

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