What is 1/2 ln(x)+8ln(x^3)-3ln(x-2) ?

The solution to 1/2 ln(x)+8ln(x^3)-3ln(x-2) is ln((x^{49/2})/((x-2)^3))

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simplify 1/2 ln(x)+8ln(x^3)-3ln(x-2)

simplify

\frac{1}{2} ln (x) + 8 ln (x^{3}) - 3 ln (x - 2)

ln (\frac{x ^{\frac{49}{2}}}{( x - 2 ) ^{3}})

Solution steps

\frac{1}{2} ln (x) + 8 ln (x^{3}) - 3 ln (x - 2)

Apply log rule:

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

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

= ln (x^{\frac{1}{2}}) + 8 ln (x^{3}) - 3 ln (x - 2)

Apply log rule:

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

8 ln (x^{3}) = ln ((x^{3})^{8})

= ln (x^{\frac{1}{2}}) + ln ((x^{3})^{8}) - 3 ln (x - 2)

Apply log rule:

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

ln (x^{\frac{1}{2}}) + ln ((x^{3})^{8}) = ln (x^{\frac{1}{2}} (x^{3})^{8})

= ln (x^{\frac{1}{2}} (x^{3})^{8}) - 3 ln (x - 2)

Simplify

x^{\frac{1}{2}} (x^{3})^{8} : x^{\frac{49}{2}}

= ln (x^{\frac{49}{2}}) - 3 ln (x - 2)

Apply log rule:

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

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

= ln (x^{\frac{49}{2}}) - ln ((x - 2)^{3})

Apply log rule:

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

ln (x^{\frac{49}{2}}) - ln ((x - 2)^{3}) = ln (\frac{x ^{\frac{49}{2}}}{( x - 2 ) ^{3}})

= ln (\frac{x ^{\frac{49}{2}}}{( x - 2 ) ^{3}})

lo g_{10} (25) + lo g_{10} (2) + lo g_{10} (0.02)

ln (\frac{0.017}{1})

ln (2 e^{2 t})

ln (\frac{1}{b})

3 ln (x + 1) + 2 ln (x - 1)

What is 1/2 ln(x)+8ln(x^3)-3ln(x-2) ?
The solution to 1/2 ln(x)+8ln(x^3)-3ln(x-2) is ln((x^{49/2})/((x-2)^3))