What is ln(|x|^{1/2}|x-6|^{1/2}) ?

The solution to ln(|x|^{1/2}|x-6|^{1/2}) is 1/2 ln|x|+1/2 ln|x-6|

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simplify ln(|x|^{1/2}|x-6|^{1/2})

simplify

ln (∣ x ∣^{\frac{1}{2}} ∣ x - 6 ∣^{\frac{1}{2}})

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 6 ∣

Solution steps

ln (∣ x ∣^{\frac{1}{2}} ∣ x - 6 ∣^{\frac{1}{2}})

Apply log rule:

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

ln (∣ x ∣^{\frac{1}{2}} ∣ x - 6 ∣^{\frac{1}{2}}) = ln (∣ x ∣^{\frac{1}{2}}) + ln (∣ x - 6 ∣^{\frac{1}{2}})

= ln (∣ x ∣^{\frac{1}{2}}) + ln (∣ x - 6 ∣^{\frac{1}{2}})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= \frac{1}{2} ln ∣ x ∣ + ln (∣ x - 6 ∣^{\frac{1}{2}})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (∣ x - 6 ∣^{\frac{1}{2}}) = \frac{1}{2} ln ∣ x - 6 ∣

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 6 ∣

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ln (10) - ln (25)

What is ln(|x|^{1/2}|x-6|^{1/2}) ?
The solution to ln(|x|^{1/2}|x-6|^{1/2}) is 1/2 ln|x|+1/2 ln|x-6|