What is 9ln(sqrt(x))-ln(sqrt((y/x)^2+1)+y/x) ?

The solution to 9ln(sqrt(x))-ln(sqrt((y/x)^2+1)+y/x) is ln(x^{7/2}(-y+sqrt(x^2+y^2)))

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simplify 9ln(sqrt(x))-ln(sqrt((y/x)^2+1)+y/x)

simplify

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

ln (x^{\frac{7}{2}} (- y + x^{2} + y^{2}))

Solution steps

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

Apply log rule:

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

9 ln (x) = ln ((x)^{9})

= ln ((x)^{9}) - ln (\frac{y}{x} + (\frac{y}{x})^{2} + 1)

Apply log rule:

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

ln ((x)^{9}) - ln ((\frac{y}{x})^{2} + 1 + \frac{y}{x}) = ln \frac{( x ) ^{9}}{\frac{y}{x} + ( \frac{y}{x} ) ^{2} + 1}

= ln \frac{( x ) ^{9}}{( \frac{y}{x} ) ^{2} + 1 + \frac{y}{x}}

Simplify

\frac{( x ) ^{9}}{( \frac{y}{x} ) ^{2} + 1 + \frac{y}{x}} : x^{\frac{7}{2}} (- y + x^{2} + y^{2})

= ln (x^{\frac{7}{2}} (- y + x^{2} + y^{2}))

2 lo g_{a} (x) - 3 lo g_{a} (y)

3 lo g_{c} (y) - \frac{15}{4} lo g_{c} (x)

\frac{1}{3} (6 ln (x + 5) + ln (x) - ln (x^{2} - 6))

1.464 - lo g_{10} (\frac{1}{1 0 ^{0.3}})

\frac{( ln ( 2 ) \cdot 45 )}{ln ( 10 ) - ln ( 2 )}

What is 9ln(sqrt(x))-ln(sqrt((y/x)^2+1)+y/x) ?
The solution to 9ln(sqrt(x))-ln(sqrt((y/x)^2+1)+y/x) is ln(x^{7/2}(-y+sqrt(x^2+y^2)))