Question 415395: What is this. Find x,y and z
log3(log4(log5x))=log4(log5(log3y))=log5(log3(log4z))=0
The numbers, log3<-...log4<-...log5<-, are bases. Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
First of all we need to understand what this "continued" equation means and how to work with them. The notation above is just a shorthand way of saying that any combination of 2 of these 4 expressions are equal to each other. There are 6 combinations of 2 of these four expressions so the notation above saves us from having to write 8 different "regular" equations (one expression equals another expression).
To work with this notation one usualy will pick two of the expressions, write a "regular" equation for those two expressions and use regular Algebra on that equation. To find x, y and z we will "extract" and use the following equations, respectively:
We use these 3 (of the six) because these are the ones that have just one variable in them.
To solve each of these equations we will "peel away" the logarithms, one-by-one, by rewriting the equaiton in exponential form repeatedly. I will do one of the three and from that you should be able to figure out the other two.
In general, the logarithmic equation is equivalent to . Using this pattern on the outermost log (the base 3 log) we get:
which simplifies to:
Repeat:
which simplifies to:
Repeat:
which simplifies to:
x = 625
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