What is simplify-3ln|x|-ln|x+2|+5ln|x-2| ?

The solution to simplify-3ln|x|-ln|x+2|+5ln|x-2| is ln((|x-2|^5)/(|x|^3|x+2|))

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simplify-3ln|x|-ln|x+2|+5ln|x-2|

simplify

- 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣

ln (\frac{∣ x - 2 ∣ ^{5}}{∣ x ∣ ^{3} ∣ x + 2 ∣})

Solution steps

- 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣

Apply log rule:

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

3 ln ∣ x ∣ = ln (∣ x ∣^{3})

= - ln (∣ x ∣^{3}) - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣

Rewrite

- ln (∣ x ∣^{3}) - ln ∣ x + 2 ∣

- (ln (∣ x ∣^{3}) + ln ∣ x + 2 ∣)

= - (ln (∣ x ∣^{3}) + ln ∣ x + 2 ∣) + 5 ln ∣ x - 2 ∣

Apply log rule:

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

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

= - ln (∣ x ∣^{3} ∣ x + 2 ∣) + 5 ln ∣ x - 2 ∣

Apply log rule:

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

5 ln ∣ x - 2 ∣ = ln (∣ x - 2 ∣^{5})

= - ln (∣ x ∣^{3} ∣ x + 2 ∣) + ln (∣ x - 2 ∣^{5})

Apply log rule:

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

ln (∣ x - 2 ∣^{5}) - ln (∣ x ∣^{3} ∣ x + 2 ∣) = ln (\frac{∣ x - 2 ∣ ^{5}}{∣ x ∣ ^{3} ∣ x + 2 ∣})

= ln (\frac{∣ x - 2 ∣ ^{5}}{∣ x ∣ ^{3} ∣ x + 2 ∣})

lo g_{10} (\frac{1}{50})

ln (8) - ln (9)

ln \frac{( x ^{4} ( x + 1 ) ^{3} )}{( x - 1 ) ^{4}}

8 lo g_{b} (x) + 9 lo g_{b} (y) - 4 lo g_{b} (z)

\frac{23}{2} (2 \cdot ln (\frac{a ^{3}}{b}) + 22 \cdot ln (\frac{1}{b}))

What is simplify-3ln|x|-ln|x+2|+5ln|x-2| ?
The solution to simplify-3ln|x|-ln|x+2|+5ln|x-2| is ln((|x-2|^5)/(|x|^3|x+2|))