What is 1/2 ln(30)+1/2 ln(1-1/2) ?

The solution to 1/2 ln(30)+1/2 ln(1-1/2) is 1/2 ln(15)

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simplify 1/2 ln(30)+1/2 ln(1-1/2)

simplify

\frac{1}{2} ln (30) + \frac{1}{2} ln (1 - \frac{1}{2})

\frac{1}{2} ln (15)

Decimal Notation

1.35402 \dots

Solution steps

\frac{1}{2} ln (30) + \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 (30) = ln (3 0^{\frac{1}{2}})

= ln (3 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 (3 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 (3 0^{\frac{1}{2}}) + ln ((1 - \frac{1}{2})^{\frac{1}{2}}) = ln (3 0^{\frac{1}{2}} (- \frac{1}{2} + 1)^{\frac{1}{2}})

= ln (3 0^{\frac{1}{2}} (- \frac{1}{2} + 1)^{\frac{1}{2}})

3 0^{\frac{1}{2}} (1 - \frac{1}{2})^{\frac{1}{2}} = 15

= ln (15)

Rewrite as

= ln (1 5^{\frac{1}{2}})

Apply log rule

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

assuming

x \geq 0

= \frac{1}{2} ln (15)

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What is 1/2 ln(30)+1/2 ln(1-1/2) ?
The solution to 1/2 ln(30)+1/2 ln(1-1/2) is 1/2 ln(15)