solve for x if 2^2x+3 =5.2^x+1

Question

2^2x+3 =5.2^x+1

Answer 1

I interpret this as 2^(2x+3)=5*2^(x+1).

Take logs of each side: (2x+3)log2=log5 + (x+1)log2.

2xlog2+3log2=log5+xlog2+log2.

xlog2=log(5*2/8)=log1.25; x=log1.25/log2=0.32193 approx.

CHECK: Put this value into the original equation: 2^(2x+3)=12.5; 2^(x+1)=2.5 so 5*2.5=12.5. The equation balances.

Answer 2

Given 2^(2x+3) =5.2^(x+1)
5.2^(x+1)=(26/5)^(x+1)
2^(2x+3)=(26/5)^(x+1)
4^x=0.65*5.2^x
x=1.64193