What is ln((16e^{-4}sqrt(2))/((x+1)^2)) ?

The solution to ln((16e^{-4}sqrt(2))/((x+1)^2)) is -2ln(x+1)+9/2 ln(2)-4

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simplify ln((16e^{-4}sqrt(2))/((x+1)^2))

simplify

ln (\frac{16 e ^{- 4} 2}{( x + 1 ) ^{2}})

- 2 ln (x + 1) + \frac{9}{2} ln (2) - 4

Solution steps

ln (\frac{16 e ^{- 4} 2}{( x + 1 ) ^{2}})

Apply log rule:

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

ln (\frac{16 e ^{- 4} 2}{( x + 1 ) ^{2}}) = ln (16 e^{- 4} 2) - ln ((x + 1)^{2})

= ln (162 e^{- 4}) - ln ((x + 1)^{2})

Apply log rule:

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

ln ((x + 1)^{2}) = 2 ln (x + 1)

= ln (16 e^{- 4} 2) - 2 ln (x + 1)

Apply log rule:

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

ln (16 e^{- 4} 2) = ln (16) + ln (e^{- 4}) + ln (2)

= ln (16) + ln (e^{- 4}) + ln (2) - 2 ln (x + 1)

Apply log rule:

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

ln (e^{- 4}) = - 4 ln (e)

= ln (16) - 4 ln (e) + ln (2) - 2 ln (x + 1)

4 ln (e) = 4

ln (2) = \frac{1}{2} ln (2)

= ln (16) - 4 + \frac{1}{2} ln (2) - 2 ln (x + 1)

ln (16) = 4 ln (2)

= 4 ln (2) - 4 + \frac{1}{2} ln (2) - 2 ln (x + 1)

Group like terms

= - 2 ln (x + 1) + 4 ln (2) + \frac{1}{2} ln (2) - 4

Add similar elements:

4 ln (2) + \frac{1}{2} ln (2) = \frac{9}{2} ln (2)

= - 2 ln (x + 1) + \frac{9}{2} ln (2) - 4

ln (\frac{x}{y ^{2} z ^{2}})

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

lo g_{10} (\frac{y}{x}) e^{4}

- lo g_{10} (1.2 \cdot 1 0^{- 6})

ln (8.2) - ln (7.5)

What is ln((16e^{-4}sqrt(2))/((x+1)^2)) ?
The solution to ln((16e^{-4}sqrt(2))/((x+1)^2)) is -2ln(x+1)+9/2 ln(2)-4