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# If $x$, $y$, and $z$ are positive real numbers satisfying:

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If x,y , and z are positive real numbers satisfying:

\begin{align*} \log x - \log y &= a \\ \log y - \log z &= 15, \text{ and} \\ \log z - \log x &= -7, \\ \end{align*}

where a is a real number, what is a?

Dec 16, 2017

#1
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log x - log y   =   a        so        x/y  =  10^a

log y - log z   =   15      so        y/z  =  10^15

log z - log x   =   -7       so        z/x  =  10^-7        so        z  =  x * 10^-7

y/(x * 10^-7)  =  10^15

y/x * 10^7  =  10^15

y/x   =   10^8

x/y   =   10^-8

10^-8  =  10^a

a  =  -8

Dec 16, 2017
#2
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log x   - log y  = a

log y  - log z  =  15

log z   - log x   =   -7

Add  the first two equations and we get that

log x  - log z   =  a  + 15

Multiplying the third equation by  -1 on both sides we have that

log x  -  log z  =  7

This implies that

a + 15  =  7         subtract 15 from both sides

a  = - 8

Dec 16, 2017
edited by CPhill  Dec 16, 2017
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thank you guys!

Dec 18, 2017