Question 202660
I'll do the first one to give you an idea on how to tackle these problems.



{{{2^(2x+1)-10*2^x=-12}}} Start with the given equation.



{{{2^(2x)*2^1-10*2^x=-12}}} Break up the exponent using the identity {{{x^(y+z)=x^(y)*x^(z)}}}



{{{2^(2x)*2-10*2^x=-12}}} Raise 2 to the first power to get 2



{{{2*2^(2x)-10*2^x=-12}}} Rearrange the terms.



{{{2*(2^x)^2-10*2^x=-12}}} Rewrite {{{2^(2x)}}} as {{{(2^x)^2}}} using the identity {{{(x^(y))^z=x^(y*z)}}}



Now let {{{z=2^x}}}. So {{{z^2=(2^x)^2}}}



{{{2z^2-10z=-12}}} Replace {{{(2^x)^2}}} with {{{z^2}}}. Replace {{{2^x}}} with {{{z}}}



{{{2z^2-10z+12=0}}} Add 12 to both sides.



Notice that the quadratic {{{2z^2-10z+12}}} is in the form of {{{Az^2+Bz+C}}} where {{{A=2}}}, {{{B=-10}}}, and {{{C=12}}}



Let's use the quadratic formula to solve for "z":



{{{z = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{z = (-(-10) +- sqrt( (-10)^2-4(2)(12) ))/(2(2))}}} Plug in  {{{A=2}}}, {{{B=-10}}}, and {{{C=12}}}



{{{z = (10 +- sqrt( (-10)^2-4(2)(12) ))/(2(2))}}} Negate {{{-10}}} to get {{{10}}}. 



{{{z = (10 +- sqrt( 100-4(2)(12) ))/(2(2))}}} Square {{{-10}}} to get {{{100}}}. 



{{{z = (10 +- sqrt( 100-96 ))/(2(2))}}} Multiply {{{4(2)(12)}}} to get {{{96}}}



{{{z = (10 +- sqrt( 4 ))/(2(2))}}} Subtract {{{96}}} from {{{100}}} to get {{{4}}}



{{{z = (10 +- sqrt( 4 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{z = (10 +- 2)/(4)}}} Take the square root of {{{4}}} to get {{{2}}}. 



{{{z = (10 + 2)/(4)}}} or {{{z = (10 - 2)/(4)}}} Break up the expression. 



{{{z = (12)/(4)}}} or {{{z =  (8)/(4)}}} Combine like terms. 



{{{z = 3}}} or {{{z = 2}}} Simplify. 



So the solutions (in terms of 'z') are {{{z = 3}}} or {{{z = 2}}}



Now recall that we let {{{z=2^x}}}. So let's use this to find the solutions in terms of "x".



{{{z = 3}}} Start with the first solution in terms of 'z'



{{{2^x = 3}}} Plug in {{{z=2^x}}}



{{{log(10,(2^x)) = log(10,(3))}}} Take the log of both sides



{{{x*log(10,(2)) = log(10,(3))}}} Rewrite the first log using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{x = log(10,(3))/log(10,(2))}}} Divide both sides by {{{log(10,(2))}}} to isolate "x"



{{{x = log(2,(3))}}} Use the change of base formula to simplify



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{{{z = 2}}}  Start with the second solution in terms of 'z'



{{{2^x = 2}}} Plug in {{{z=2^x}}}



{{{log(10,(2^x)) = log(10,(2))}}} Take the log of both sides



{{{x*log(10,(2)) = log(10,(2))}}} Rewrite the first log using the identity  {{{log(b,(x^y))=y*log(b,(x))}}}



{{{x = log(10,(2))/log(10,(2))}}} Divide both sides by {{{log(10,(2))}}} to isolate "x"



{{{x = log(2,(2))}}} Use the change of base formula to simplify



{{{x = 1}}} Evaluate the log base 2 of 2 to get 1



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Answer:



So the solutions are {{{x = log(2,(3))}}} or {{{x = 1}}}