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Solve the equation log(1//3)[2(1/2)^(x)-...

Solve the equation `log_(1//3)[2(1/2)^(x)-1]=log_(1//3)[(1/4)^(x)-4]`

Text Solution

Verified by Experts

The given equation is equivalent to
`{(2(1/2)^(x)-1gt0),(2(1/2)^(x)-1=(1/4)^(x)-4):}`
`implies{((1/2)^(x)gt1/2),((1/2)^(2x)-2(1/2)^(x)-3=0):}`
`implies{(xlt1),([(1/2)^(x)-3][(1/2)^(x)+1]):}=0`
`implies{(xlt1),((1/2)^(x)=3,(1/2)^(x)+1!=0):}implies{(xlt1),(x=(-log_(2)3)):}`
Hence `x_(1)=-log_(2)3` is the root of the original equation.
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