`log(8x) - log(1 + sqrt(x)) = 2` Solve the logarithmic equation algebraically. Approximate the result to three decimal places.

Textbook Question

Chapter 3, 3.4 - Problem 62 - Precalculus (3rd Edition, Ron Larson).
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kspcr111 | In Training Educator

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Given

`log(8x) - log(1 + sqrt(x)) = 2` ---------------------(1)

On simplification we get

=> As we know `log(a) - log(b) = log(a/b)` ` `

so ,

=> ` log(8x) - log(1 + sqrt(x)) = 2` ` `

=> `log((8x)/(1 + sqrt(x))) = 2` ` `

=>` `   `log((8x)/(1 + sqrt(x))) = log 10^2`    [as `2= log_10 (10^2) ` ` ` ]

=> removing log on both sides we get

=>`(8x)/(1 + sqrt(x)) = 10 ^2` ` `

=>`8x = 100(1+sqrt(x))` ` `

=>`(8x - 100 ) = 100 sqrt(x)`

=> squaring on both sides we get

=> `(8x-100)^2 = 10000x`

=>`64x^2 +10000 -1600x = 10000x` ` `

=>`64x^2 +10000 - 11600x = 0` ` `

=>on simplification we get

`x= +- (25/8)(29+5sqrt(33))`

` `

on verification by substituting the values of x in the equation (1)

  we get `x= + (25/8)(29+5sqrt(33))` ` `

as ` ` cannot be solved `x= - (25/8)(29+5sqrt(33))`   when  "sqrt(x)"

so,

x=  (25/8)(29+5sqrt(33))  

=>  x` ` = 180.384

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