What is 1/2 ln|x-3|+1/2 ln|x+3|-ln|x| ?

The solution to 1/2 ln|x-3|+1/2 ln|x+3|-ln|x| is ln((|x-3|^{1/2}|x+3|^{1/2})/(|x|))

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simplify 1/2 ln|x-3|+1/2 ln|x+3|-ln|x|

simplify

\frac{1}{2} ln ∣ x - 3 ∣ + \frac{1}{2} ln ∣ x + 3 ∣ - ln ∣ x ∣

ln (\frac{∣ x - 3 ∣ ^{\frac{1}{2}} ∣ x + 3 ∣ ^{\frac{1}{2}}}{∣ x ∣})

Solution steps

\frac{1}{2} ln ∣ x - 3 ∣ + \frac{1}{2} ln ∣ x + 3 ∣ - ln ∣ x ∣

Apply log rule:

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

\frac{1}{2} ln ∣ x - 3 ∣ = ln (∣ x - 3 ∣^{\frac{1}{2}})

= ln (∣ x - 3 ∣^{\frac{1}{2}}) + \frac{1}{2} ln ∣ x + 3 ∣ - ln ∣ x ∣

Apply log rule:

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

\frac{1}{2} ln ∣ x + 3 ∣ = ln (∣ x + 3 ∣^{\frac{1}{2}})

= ln (∣ x - 3 ∣^{\frac{1}{2}}) + ln (∣ x + 3 ∣^{\frac{1}{2}}) - ln ∣ x ∣

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (∣ x - 3 ∣^{\frac{1}{2}}) + ln (∣ x + 3 ∣^{\frac{1}{2}}) = ln (∣ x - 3 ∣^{\frac{1}{2}} ∣ x + 3 ∣^{\frac{1}{2}})

= ln (∣ x - 3 ∣^{\frac{1}{2}} ∣ x + 3 ∣^{\frac{1}{2}}) - ln ∣ x ∣

Apply log rule:

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

ln (∣ x - 3 ∣^{\frac{1}{2}} ∣ x + 3 ∣^{\frac{1}{2}}) - ln ∣ x ∣ = ln (\frac{∣ x - 3 ∣ ^{\frac{1}{2}} ∣ x + 3 ∣ ^{\frac{1}{2}}}{∣ x ∣})

= ln (\frac{∣ x - 3 ∣ ^{\frac{1}{2}} ∣ x + 3 ∣ ^{\frac{1}{2}}}{∣ x ∣})

ln (\frac{1}{0.15})

ln (\frac{84}{92})

\frac{1}{0.26 - 0.17} ln (\frac{0.26}{0.17})

\frac{1}{5} ln (\frac{1}{5 - 0.8 Q})

ln (\frac{100}{200})

What is 1/2 ln|x-3|+1/2 ln|x+3|-ln|x| ?
The solution to 1/2 ln|x-3|+1/2 ln|x+3|-ln|x| is ln((|x-3|^{1/2}|x+3|^{1/2})/(|x|))