What is ln(12)+1/2 ln(x/2)+1/2 ln(x/2) ?

The solution to ln(12)+1/2 ln(x/2)+1/2 ln(x/2) is ln(6x)

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simplify ln(12)+1/2 ln(x/2)+1/2 ln(x/2)

simplify

ln (12) + \frac{1}{2} ln (\frac{x}{2}) + \frac{1}{2} ln (\frac{x}{2})

ln (6 x)

Solution steps

ln (12) + \frac{1}{2} ln (\frac{x}{2}) + \frac{1}{2} ln (\frac{x}{2})

Apply log rule:

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

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

= ln (12) + ln ((\frac{x}{2})^{\frac{1}{2}}) + \frac{1}{2} ln (\frac{x}{2})

Apply log rule:

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

ln (12) + ln ((\frac{x}{2})^{\frac{1}{2}}) = ln (12 (\frac{x}{2})^{\frac{1}{2}})

= ln (12 (\frac{x}{2})^{\frac{1}{2}}) + \frac{1}{2} ln (\frac{x}{2})

Simplify

12 (\frac{x}{2})^{\frac{1}{2}} : 62 x

= ln (62 x) + \frac{1}{2} ln (\frac{x}{2})

Apply log rule:

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

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

= ln (62 x) + ln ((\frac{x}{2})^{\frac{1}{2}})

Apply log rule:

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

ln (62 x) + ln ((\frac{x}{2})^{\frac{1}{2}}) = ln (62 (\frac{x}{2})^{\frac{1}{2}} x)

= ln (62 (\frac{x}{2})^{\frac{1}{2}} x)

Simplify

62 x (\frac{x}{2})^{\frac{1}{2}} : 6 x

= ln (6 x)

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What is ln(12)+1/2 ln(x/2)+1/2 ln(x/2) ?
The solution to ln(12)+1/2 ln(x/2)+1/2 ln(x/2) is ln(6x)