Find the exact value of x Logarithms

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To find the exact value of x in the equation (3x)lg3=(4x)lg4, the logarithmic properties are applied. The equation is manipulated by expressing both sides as powers of 10 and comparing exponents. This leads to isolating lg(x) and simplifying using the difference of squares identity. The final solution reveals that x equals 1/12. The discussion emphasizes the importance of logarithmic manipulation in solving equations.
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Homework Statement



Find the exact value of x if:

Homework Equations


(3x)lg3=(4x)lg4.


The Attempt at a Solution


3lg3xlg3=4lg4xlg4
(xlg3)/(xlg4)=(4lg4)/(3lg3)
xlg3-lg4=(4lg4)(3-(lg3))
xlg(3/4)=(4lg4)(3lg(1/3))

Please help me, I am stuck here!


 
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Write both sides of the original equation as power of 10, and compare the exponents (which is the same as taking the lg of both sides)
So you get

lg(3)lg(3x)=lg(4)lg(4x)

that is, (lg3)2+(lg3) lg(x)=(lg4)2+(lg4)lg(x)

Isolate lg(x), use the identity a2-b2=(a-b)(a+b) and simplify. You get lg(x) as the logarithm of a number, from which the exact value of x is found.

ehild
 
Last edited:


If xa=b, then x=b(1/a).
 


Thank you so much echild. The answer is (1/12)
 


Exactly! Well done!

ehild
 

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