What is 1/2 log_{e}(1+(2y)/x)+log_{e}(x) ?

The solution to 1/2 log_{e}(1+(2y)/x)+log_{e}(x) is ln(((x+2y)/x)^{1/2}x)

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simplify 1/2 log_{e}(1+(2y)/x)+log_{e}(x)

simplify

\frac{1}{2} lo g_{e} (1 + \frac{2 y}{x}) + lo g_{e} (x)

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

Solution steps

\frac{1}{2} lo g_{e} (1 + \frac{2 y}{x}) + lo g_{e} (x)

Apply log rule:

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

\frac{1}{2} lo g_{e} (\frac{2 y}{x} + 1) = lo g_{e} ((\frac{2 y}{x} + 1)^{\frac{1}{2}})

= lo g_{e} ((\frac{2 y}{x} + 1)^{\frac{1}{2}}) + lo g_{e} (x)

Apply log rule:

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

lo g_{e} ((1 + \frac{2 y}{x})^{\frac{1}{2}}) + lo g_{e} (x) = lo g_{e} (x (\frac{2 y}{x} + 1)^{\frac{1}{2}})

= lo g_{e} (x (\frac{2 y}{x} + 1)^{\frac{1}{2}})

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

= lo g_{e} ((\frac{x + 2 y}{x})^{\frac{1}{2}} x)

Apply log rule:

lo g_{e} (x) = ln (x)

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

\frac{1}{2} lo g_{10} (7)

\frac{d}{d x} (\frac{1}{8} ln (4 ∣ ln (x) ∣)) + c

5 ln (x) + \frac{3}{4} ln (y)

- ln (2 - y) - ln (x)

ln (e^{6} 4 - x)

What is 1/2 log_{e}(1+(2y)/x)+log_{e}(x) ?
The solution to 1/2 log_{e}(1+(2y)/x)+log_{e}(x) is ln(((x+2y)/x)^{1/2}x)