What is ln(4xe^{-20x}) ?

The solution to ln(4xe^{-20x}) is 2ln(2)+ln(x)-20x

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simplify ln(4xe^{-20x})

simplify

ln (4 x e^{- 20 x})

2 ln (2) + ln (x) - 20 x

Solution steps

ln (4 x e^{- 20 x})

Apply log rule:

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

ln (4 x e^{- 20 x}) = ln (4) + ln (x) + ln (e^{- 20 x})

= ln (4) + ln (x) + ln (e^{- 20 x})

Apply log rule:

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

ln (e^{- 20 x}) = - 20 x ln (e)

= ln (4) + ln (x) - 20 x ln (e)

20 x ln (e) = 20 x

= ln (4) + ln (x) - 20 x

ln (4) = 2 ln (2)

= 2 ln (2) + ln (x) - 20 x

10 lo g_{10} (\frac{( 2.5 \cdot 1 0 ^{- 3} )}{1000}) + 40 + 59

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

ln (10000) - ln (5000) + 5

ln (\frac{24}{5})

+ \frac{ln ( 1.5575 )}{ln ( \frac{2}{25} ) - ln ( 0.239 )}