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

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

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

simplify

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

ln (1 + e^{x + 2 y}) - x - 2 y

Solution steps

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

Apply log rule:

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

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

= ln (e^{x + 2 y} + 1) - ln (e^{x + 2 y})

Apply log rule:

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

ln (e^{x + 2 y}) = (x + 2 y) ln (e)

= ln (1 + e^{x + 2 y}) - (x + 2 y) ln (e)

(x + 2 y) ln (e) = x + 2 y

= ln (e^{x + 2 y} + 1) - (x + 2 y)

- (x + 2 y) : - x - 2 y

= ln (1 + e^{x + 2 y}) - x - 2 y

\frac{1}{3} ln (\frac{9 x ^{2} - 16 + 3 x}{4}) + C

ln (20000 \cdot 10000)

ln (294) - ln (51.5)

2 [2 ln (x) - ln (x + 4) - ln (x - 4)]

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

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