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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{∣ z + 4 ∣ ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣ ^{\frac{6}{5}}} : \frac{∣ z + 4 ∣ ^{\frac{4}{5}} ∣ z - 1 ∣ ^{\frac{6}{5}}}{∣ z + 4 ∣}

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

Apply log rule:

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

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

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

Simplify

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

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

\frac{1}{2} ln ∣ 2 (10.3) + 1 ∣ + 2 ln ∣ (10.3) - 1 ∣

\frac{1}{4} ln (\frac{3 e ^{\frac{3}{2} x}}{3 e ^{\frac{3}{2} x} - 2})

ln (\frac{2.3}{2.6})

\frac{1}{2} [ln (x + 1) + 2 ln (x - 1)] + 3 ln (x)

- 0.3 lo g_{2} (0.3) - 0.7 lo g_{2} (0.7)

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