To silve 2^(2x-1) = 8^x
The LHS has a base 2 and the RHS has a base 8.
We can have acommon base and then equate the powers.
We know that 2 = 8^(1/3). So we convert the LHS to the base 8.
LHS = 2^(2x-1) =(8^(1/3)) ^(2x-1) = 8^((2x-1)/3) = RHS = 8^x , (as a^m)^n = a^(mn). Now the original equation is rewritten as:
8^((2x-1)/3) = 8^x. Nowboth sides have the same base 8and we can equate the powers on both sides.
(2x-1)/3 = x. Or
2x-1 = 3x.
2x-3x = 1.
-x = 1
x = -1
Let us check: 2^(2x-1) = 8^x . Put x = -1, then
LHS : 2^(2x-1)2^2 = (-2*1-1) = 2^-3 =1/8
RHS: 8^x= 8^(-1) = 1/8.
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