What is 1/200 ln|x|-1/200 ln|x-200| ?

The solution to 1/200 ln|x|-1/200 ln|x-200| is ln(\sqrt[200]{(|x|)/(|x-200|)})

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simplify 1/200 ln|x|-1/200 ln|x-200|

simplify

\frac{1}{200} ln ∣ x ∣ - \frac{1}{200} ln ∣ x - 200 ∣

Solution steps

\frac{1}{200} ln ∣ x ∣ - \frac{1}{200} ln ∣ x - 200 ∣

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{200} ln ∣ x ∣ = ln (∣ x ∣^{\frac{1}{200}})

= ln (∣ x ∣^{\frac{1}{200}}) - \frac{1}{200} ln ∣ x - 200 ∣

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{200} ln ∣ x - 200 ∣ = ln (∣ x - 200 ∣^{\frac{1}{200}})

= ln (∣ x ∣^{\frac{1}{200}}) - ln (∣ x - 200 ∣^{\frac{1}{200}})

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (∣ x ∣^{\frac{1}{200}}) - ln (∣ x - 200 ∣^{\frac{1}{200}}) = ln (\frac{∣ x ∣ ^{\frac{1}{200}}}{∣ x - 200 ∣ ^{\frac{1}{200}}})

= ln (\frac{∣ x ∣ ^{\frac{1}{200}}}{∣ x - 200 ∣ ^{\frac{1}{200}}})

Combine same powers :

480 ln (\frac{A}{275})

2 ln ∣ x - 2 ∣ - 3 ln ∣ x - 3 ∣

ln (x^{4} - 4) - ln (x^{2})

lo g_{7} (x (x - 10))

2 lo g_{x} (- 3 lo g_{10} (x + 1))

What is 1/200 ln|x|-1/200 ln|x-200| ?
The solution to 1/200 ln|x|-1/200 ln|x-200| is ln(\sqrt[200]{(|x|)/(|x-200|)})