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#