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

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Answer 1 · 2016-07-04T09:48:38

#x=- (2ln(5))/(ln(2)-ln(5)) ~~+3.513# to 3 decimal places

#color(brown)("Take logs of both sides")#

#" "ln(2^x)=ln(5^(x-2))#

#color(brown)("This is the same as")#

#" "xln(2)=(x-2)ln(5)#

#color(brown)("Divide both sides by "ln(2))#

#" "x=(x-2)(ln(5))/ln(2)#

#color(brown)("Multiply out the bracket")#

#" "x=xln(5)/ln(2)-(2ln(5))/ln(2)#

#color(brown)("Subtract "xln(5)/ln(2)" from both sides")#

#" "x-xln(5)/ln(2)=-(2ln(5))/(ln(2))#

#color(brown)("Factor out the "x)#

#" "x(1-ln(5)/ln(2))=-(2ln(5))/ln(2)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(green)("Note that "(1-ln(5)/ln(2)) = (ln(2)-ln(5))/ln(2))#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Divide both sides by "(ln(2)-ln(5))/ln(2))#

#" "x=-(2ln(5))/(cancel(ln(2)))xx(cancel(ln(2)))/(ln(2)-ln(5))#

#" "x=- (2ln(5))/(ln(2)-ln(5)) ~~+3.513# to 3 decimal places