Determine x if (lg x)^2 = lg 100x?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to convert the logarithm of the product to the right into a sum of logarithms, such that:

`log (100 x) = log 100 + log x`

`log (100 x) = log 10^2 + log x`

`log (100 x) = 2log 10 + log x`

Since `log 10 = 1` , yields:

`log (100 x) = 2 + log x`

You need to perform the following substitution, such that:

`log x = y`

Replacing the variable, yields:

`y^2 = 2 + y`

Moving all terms to one side, yields:

`y^2 - y - 2 = 0`

Using quadratic formula, yields:

`y_(1,2) = (1+-sqrt(1+8))/2 => y_(1,2) = (1+-3)/2`

`y_1 = 2; y_2 = -1`

You need to solve the following equations, such that:

`log x = y_1 => log x = 2 => x = 10^2 => x = 100`

`log x = y_2 => log x = -1 => x = 10^(-1) => x = 1/10`

Hence, evaluating the solutions to the given equation, yields `x = 1/10, x = 100.`

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