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# 4^(x-1) = 2^x + 8

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4^(x-1) = 2^x + 8

Guest May 30, 2015

#1
+17711
+10

Since 4 = 22   --->   4x-1  =  (22)x-1  =  22x-2

Since 22x-2  =  22x · 2-2  =  22x · ¼  =  ¼ · 22x

4x-1  =  2x + 8   --->   ¼ · 22x  =  2x + 8    --->   ¼ · 22x - 2x - 8 = 0

Multiply by 4:   --->  22x - 4 · 2x - 32 = 0

Since  22x  =  (2x)2     --->   22x - 4 · 2x - 32 = 0   --->  (2x)- 4 · 2x - 32 = 0

Since A2 - 4A - 32  =  0  factors into (A - 8)(A + 4)  =  0

Substitute 2x for A:

[ 2x - 8 ] [ 2x + 4 ]  =  0

So, either 2x - 8  =  0     or     2x + 4  =  0

--->   2x - 8  =  0   --->   2x  =  8   --->   x  =  3    <--- Answer!

--->   2x + 4  =  0   will not give a real answer.

geno3141  May 31, 2015
Sort:

#1
+17711
+10

Since 4 = 22   --->   4x-1  =  (22)x-1  =  22x-2

Since 22x-2  =  22x · 2-2  =  22x · ¼  =  ¼ · 22x

4x-1  =  2x + 8   --->   ¼ · 22x  =  2x + 8    --->   ¼ · 22x - 2x - 8 = 0

Multiply by 4:   --->  22x - 4 · 2x - 32 = 0

Since  22x  =  (2x)2     --->   22x - 4 · 2x - 32 = 0   --->  (2x)- 4 · 2x - 32 = 0

Since A2 - 4A - 32  =  0  factors into (A - 8)(A + 4)  =  0

Substitute 2x for A:

[ 2x - 8 ] [ 2x + 4 ]  =  0

So, either 2x - 8  =  0     or     2x + 4  =  0

--->   2x - 8  =  0   --->   2x  =  8   --->   x  =  3    <--- Answer!

--->   2x + 4  =  0   will not give a real answer.

geno3141  May 31, 2015

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