What is ln(2+e^1)-ln(2+e^{-3}) ?

The solution to ln(2+e^1)-ln(2+e^{-3}) is ln((2e^3+e^4)/(2e^3+1))

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simplify ln(2+e^1)-ln(2+e^{-3})

simplify

ln (2 + e^{1}) - ln (2 + e^{- 3})

ln (\frac{2 e ^{3} + e ^{4}}{2 e ^{3} + 1})

Decimal Notation

0.83370 \dots

Solution steps

ln (2 + e^{1}) - ln (2 + e^{- 3})

Apply log rule:

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

ln (2 + e^{1}) - ln (2 + e^{- 3}) = ln (\frac{e ^{1} + 2}{e ^{- 3} + 2})

= ln (\frac{2 + e ^{1}}{2 + e ^{- 3}})

\frac{2 + e ^{1}}{2 + e ^{- 3}} = \frac{2 e ^{3} + e ^{4}}{2 e ^{3} + 1}

= ln (\frac{2 e ^{3} + e ^{4}}{2 e ^{3} + 1})

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

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

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

lo g_{2} (x) - 5 lo g_{2} (y)

lo g_{5} (x + 12) + lo g_{5} (x - 12)

What is ln(2+e^1)-ln(2+e^{-3}) ?
The solution to ln(2+e^1)-ln(2+e^{-3}) is ln((2e^3+e^4)/(2e^3+1))