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March 13th, 2016, 01:50 PM   #1
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Find log x in terms of a and b.

If $\log(x^3 y^2)=a$ and $\log (\frac{y}{x})=b$ find log x in terms of a and b. How do I go about this?

Last edited by skipjack; March 16th, 2016 at 02:03 PM.
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March 13th, 2016, 02:00 PM   #2
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If log $(x^3 y^2)=a$ and $log (\frac{y}{x})=b$ find log x in terms of a and b. How do I go about this?
laws of logarithms ...

$3\log{x} + 2\log{y} = a$

$\log{y}-\log{x} = b$

multiply each term of the second equation by -2 and add to the first equation ...

$3\log{x} + 2\log{y} = a$
$-2\log{y}+2\log{x} = -2b$
-----------------------------
$5\log{x} = a-2b$

$\log{x} = \dfrac{a-2b}{5}$
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March 16th, 2016, 01:11 PM   #3
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But without adding the two equations to eliminate logy, how else can this be done?
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March 16th, 2016, 01:31 PM   #4
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But without adding the two equations to eliminate logy, how else can this be done?
since $\log{y}-\log{x}=b$, then $\log{y}=b+\log{x}$

substitute $(b+\log{x})$ in for $\log{y}$ in the first equation ...

$3\log{x}+2(b+\log{x}) = a$

$3\log{x} + 2\log{x} + 2b = a$

$5\log{x} = a-2b$

$\log{x} = \dfrac{a-2b}{5}$
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March 16th, 2016, 08:52 PM   #5
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Alright! Thank you. Apart from elimination and substitution method, any other method?
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March 17th, 2016, 05:06 AM   #6
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What method are you looking for ... ?
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March 17th, 2016, 07:46 AM   #7
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No other method.
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March 17th, 2016, 09:48 AM   #8
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Alright! Thank you. Apart from elimination and substitution method, any other method?
Graph? Echelon?
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