What is the answer to 32+2^{x-1}=2^x ?

The answer to 32+2^{x-1}=2^x is x=6

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32+2^{x-1}=2^x

32 + 2^{x - 1} = 2^{x}

x = 6

Solution steps

32 + 2^{x - 1} = 2^{x}

Subtract

2^{x}

from both sides

32 + 2^{x - 1} - 2^{x} = 2^{x} - 2^{x}

Simplify

32 + 2^{x - 1} - 2^{x} = 0

Factor

32 + 2^{x - 1} - 2^{x} : \frac{1}{2} (64 - 2^{x})

\frac{1}{2} (64 - 2^{x}) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

64 - 2^{x} = 0

Solve

64 - 2^{x} = 0 : x = 6

x = 6

e^{2 x + 1} = 9 e^{1 - x}

2^{4 x} - 2^{2 x} - 12 = 0

3^{2 x - 1} = 9^{x - 2}

133.9 = \frac{133.9}{1 + 2 e ^{- x}}

3 e^{- 5 x - 8} - 4 = 9