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)/(x^{10)|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}}{x ^{10} ∣ 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} (\frac{a}{b})

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

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

Simplify

\frac{\frac{∣ x ∣ ( y - x ) ^{6}}{x ^{6} ∣ y + 4 x ∣}}{∣ x ∣ ^{5}} : \frac{( y - x ) ^{6}}{x ^{10} ∣ y + 4 x ∣}

= ln (\frac{( y - x ) ^{6}}{x ^{10} ∣ y + 4 x ∣})

3727.8 \cdot ln (\frac{( 23180 + 56088 )}{23180})

\frac{1}{3} ln (x) + \frac{1}{3} ln (3 x)

2 ln (3) - 4 ln (7)

\frac{( - 12 [ ln ( 9.5 + 2 x ) - ln ( 6.5 + 2 x ) ] ) ^{'}}{4}

lo g_{b} (8) + lo g_{b} (3)

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)/(x^{10)|y+4x|})