What is 1/2 ln(40)+1/2 ln(1-1/2) ?

The solution to 1/2 ln(40)+1/2 ln(1-1/2) is ln(2sqrt(5))

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simplify 1/2 ln(40)+1/2 ln(1-1/2)

simplify

\frac{1}{2} ln (40) + \frac{1}{2} ln (1 - \frac{1}{2})

ln (25)

Decimal Notation

1.49786 \dots

Solution steps

\frac{1}{2} ln (40) + \frac{1}{2} ln (1 - \frac{1}{2})

Apply log rule:

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

\frac{1}{2} ln (40) = ln (4 0^{\frac{1}{2}})

= ln (4 0^{\frac{1}{2}}) + \frac{1}{2} ln (- \frac{1}{2} + 1)

Apply log rule:

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

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

= ln (4 0^{\frac{1}{2}}) + ln ((- \frac{1}{2} + 1)^{\frac{1}{2}})

Apply log rule:

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

ln (4 0^{\frac{1}{2}}) + ln ((1 - \frac{1}{2})^{\frac{1}{2}}) = ln (4 0^{\frac{1}{2}} (- \frac{1}{2} + 1)^{\frac{1}{2}})

= ln (4 0^{\frac{1}{2}} (- \frac{1}{2} + 1)^{\frac{1}{2}})

4 0^{\frac{1}{2}} (1 - \frac{1}{2})^{\frac{1}{2}} = 25

= ln (25)

\frac{1}{3} (lo g_{a} (2 x^{2}) - lo g_{a} (b y^{5}))

ln (ib)

- lo g_{10} (4.56 \cdot 1 0^{- 6})

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

- 98.97 \cdot ln (\frac{0.1}{10})

What is 1/2 ln(40)+1/2 ln(1-1/2) ?
The solution to 1/2 ln(40)+1/2 ln(1-1/2) is ln(2sqrt(5))