Question: "Solve the exponential equation by expressing each side..."

Cheyanne Leigh

Answered question

2021-09-11

Solve the exponential equation by expressing each side as a power of the same base and then equating exponents.

4^{x} = \frac{1}{\sqrt{2}}

dieseisB

Skilled2021-09-12Added 85 answers

Given:

4^{x} = \frac{1}{\sqrt{2}}

We have to expressing each side as a power of the same base
We have

4^{x} = \frac{1}{\sqrt{2}}

Consider the Right-hand side of the equation

\frac{1}{\sqrt{2}} = {(\sqrt{2})}^{- 1}

but

\sqrt{2} = 2^{\frac{1}{2}}

\frac{1}{\sqrt{2}} = {({(2)}^{(\frac{1}{2})})}^{(- 1)} = {(2)}^{\frac{1}{2} (- 1)}

= 2^{\frac{- 1}{2}}

Consider the Left-hand side of the equation

4^{x} = (2^{2}) x = 22 x

Hence the expression of each side as a power of the same base which is 2.
Now equating exponent :

2^{2 x} = 2^{\frac{- 1}{2}}

we write

2 x = - \frac{1}{2}

x = - \frac{1}{4} = - 0.25

Hence

x = - 0.25

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