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

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

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

simplify

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

ln (\frac{( y - x ) ^{6}}{∣ y + 4 x ∣})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (\frac{y}{x} - 1^{6}) - ln \frac{y}{x} + 4 = ln (\frac{\frac{y}{x} - 1 ^{6}}{\frac{y}{x} + 4})

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

Simplify

\frac{\frac{y}{x} - 1 ^{6}}{\frac{y}{x} + 4} : \frac{∣ x ∣ ( y - x ) ^{6}}{x ^{6} ∣ y + 4 x ∣}

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

Apply log rule:

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

5 ln ∣ x ∣ = ln (∣ x ∣^{5})

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

Apply log rule:

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

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

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

Simplify

\frac{∣ x ∣ ( y - x ) ^{6}}{x ^{6} ∣ y + 4 x ∣} ∣ x ∣^{5} : \frac{( y - x ) ^{6}}{∣ y + 4 x ∣}

= ln (\frac{( y - x ) ^{6}}{∣ y + 4 x ∣})

ln (\frac{x ^{4}}{e ^{3 y}})

3 ln (3) - 2 ln (5) - ln (7)

\frac{1}{2} ln (x + 5) - 2 ln (x) + ln (y)

- lo g_{e} (x) - lo g_{e} (y)

- 3 ln (x) + x + x y + y - 3 ln (y)

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