What is 6/5 ln|z-1|-6/5 ln|z+4|+ln|z+4|+ln|x| ?

The solution to 6/5 ln|z-1|-6/5 ln|z+4|+ln|z+4|+ln|x| is ln(\sqrt[5]{((z-1)^6)/((z+4)^6)}|z+4||x|)

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simplify 6/5 ln|z-1|-6/5 ln|z+4|+ln|z+4|+ln|x|

simplify

\frac{6}{5} ln ∣ z - 1 ∣ - \frac{6}{5} ln ∣ z + 4 ∣ + ln ∣ z + 4 ∣ + ln ∣ x ∣

Solution steps

\frac{6}{5} ln ∣ z - 1 ∣ - \frac{6}{5} ln ∣ z + 4 ∣ + ln ∣ z + 4 ∣ + ln ∣ x ∣

Apply log rule:

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

\frac{6}{5} ln ∣ z - 1 ∣ = ln (∣ z - 1 ∣^{\frac{6}{5}})

= ln (∣ z - 1 ∣^{\frac{6}{5}}) - \frac{6}{5} ln ∣ z + 4 ∣ + ln ∣ z + 4 ∣ + ln ∣ x ∣

Apply log rule:

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

\frac{6}{5} ln ∣ z + 4 ∣ = ln (∣ z + 4 ∣^{\frac{6}{5}})

= ln (∣ z - 1 ∣^{\frac{6}{5}}) - ln (∣ z + 4 ∣^{\frac{6}{5}}) + ln ∣ z + 4 ∣ + ln ∣ x ∣

Apply log rule:

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

ln (∣ z - 1 ∣^{\frac{6}{5}}) - ln (∣ z + 4 ∣^{\frac{6}{5}}) = ln (\frac{∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣ ^{\frac{6}{5}}})

= ln (\frac{∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣ ^{\frac{6}{5}}}) + ln ∣ z + 4 ∣ + ln ∣ x ∣

Simplify

Apply log rule:

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

Apply log rule:

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

- lo g_{10} (6.64 \cdot 1 0^{- 3})

5 + 2 ln (2) - 2 ln (5)

\frac{- 3 ln ( 3 ) - 6 ln ( 5 )}{ln ( 5 ) - ln ( 3 )}

ln (z^{2} e^{4 z})

ln (x) - ln (x - y)

What is 6/5 ln|z-1|-6/5 ln|z+4|+ln|z+4|+ln|x| ?
The solution to 6/5 ln|z-1|-6/5 ln|z+4|+ln|z+4|+ln|x| is ln(\sqrt[5]{((z-1)^6)/((z+4)^6)}|z+4||x|)