What is 1/2 ln(5x^2)-1/2 ln(2x^2) ?

The solution to 1/2 ln(5x^2)-1/2 ln(2x^2) is 1/2 ln(5/2)

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simplify 1/2 ln(5x^2)-1/2 ln(2x^2)

simplify

\frac{1}{2} ln (5 x^{2}) - \frac{1}{2} ln (2 x^{2})

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

Decimal Notation

0.45814 \dots

Solution steps

\frac{1}{2} ln (5 x^{2}) - \frac{1}{2} ln (2 x^{2})

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

Rewrite as

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

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What is 1/2 ln(5x^2)-1/2 ln(2x^2) ?
The solution to 1/2 ln(5x^2)-1/2 ln(2x^2) is 1/2 ln(5/2)