How do you solve $2^x = 5^(x - 2)$ ?

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Answer 1 · 2016-04-10T20:59:11

I found: $x=(2ln(5))/((ln(5)-ln(2)))=3.513$

I would take the natural log of both sides:

$color(red)(ln)2^x=color(red)(ln)5^(x-2)$

then use the fact that $logx^m=mlogx$ and write:
$xln(2)=(x-2)ln(5)$

rearrange:

$xln(2)-xln(5)=-2ln(5)$
$x[ln(2)-ln(5)]=-2ln(5)$

and:

$x=(-2ln(5))/((ln(2)-ln(5)))=(2ln(5))/((ln(5)-ln(2)))=3.513$

How do you solve 2^x = 5^(x - 2)2^x = 5^(x - 2)?