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Popular >

simplify 2ln(1/2)-2ln(3/2)

Solution

simplify

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

Solution

- 2 ln (3)

+1

Decimal Notation

- 2.19722 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

2 ln (\frac{3}{2}) = ln ((\frac{3}{2})^{2})

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

Apply log rule:

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

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

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

\frac{( \frac{1}{2} ) ^{2}}{( \frac{3}{2} ) ^{2}} = \frac{1}{9}

= ln (\frac{1}{9})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (9)

Rewrite

9

in power-base form:

9 = 3^{2}

= - ln (3^{2})

Apply log rule:

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

ln (3^{2}) = 2 ln (3)

= - 2 ln (3)

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Frequently Asked Questions (FAQ)

What is 2ln(1/2)-2ln(3/2) ?
The solution to 2ln(1/2)-2ln(3/2) is -2ln(3)