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

The solution to ln(x(sqrt(y/x+1)+1)^2) is ln(x)+2ln(sqrt((y+x)/x)+1)

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

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

Join

\frac{y}{x} + 1 : \frac{y + x}{x}

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

- ln (cos (x)) + ln (cos (y))

\frac{1}{3} ln (16 p) - 3 ln (2 pq) - 2 ln (5 p^{4}) + 6 ln (q)

ln (17) - 2 ln (x) + 3 ln (y)

\frac{1}{2} (ln (x) - ln (y)) + 3 ln (z)

ln (\frac{15}{22})

What is ln(x(sqrt(y/x+1)+1)^2) ?
The solution to ln(x(sqrt(y/x+1)+1)^2) is ln(x)+2ln(sqrt((y+x)/x)+1)