What is 2ln(x)+1/2 ln(x^2-4)-3ln(1/(x^2+2)) ?

The solution to 2ln(x)+1/2 ln(x^2-4)-3ln(1/(x^2+2)) is ln(x^2(x^2+2)^3(x^2-4)^{1/2})

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simplify 2ln(x)+1/2 ln(x^2-4)-3ln(1/(x^2+2))

simplify

2 ln (x) + \frac{1}{2} ln (x^{2} - 4) - 3 ln (\frac{1}{x ^{2} + 2})

ln (x^{2} (x^{2} + 2)^{3} (x^{2} - 4)^{\frac{1}{2}})

Solution steps

2 ln (x) + \frac{1}{2} ln (x^{2} - 4) - 3 ln (\frac{1}{x ^{2} + 2})

Apply log rule:

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

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

= ln (x^{2}) + \frac{1}{2} ln (x^{2} - 4) - 3 ln (\frac{1}{x ^{2} + 2})

Apply log rule:

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

\frac{1}{2} ln (x^{2} - 4) = ln ((x^{2} - 4)^{\frac{1}{2}})

= ln (x^{2}) + ln ((x^{2} - 4)^{\frac{1}{2}}) - 3 ln (\frac{1}{x ^{2} + 2})

Apply log rule:

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

ln (x^{2}) + ln ((x^{2} - 4)^{\frac{1}{2}}) = ln (x^{2} (x^{2} - 4)^{\frac{1}{2}})

= ln (x^{2} (x^{2} - 4)^{\frac{1}{2}}) - 3 ln (\frac{1}{x ^{2} + 2})

Apply log rule:

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

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

= ln (x^{2} (x^{2} - 4)^{\frac{1}{2}}) - ln ((\frac{1}{x ^{2} + 2})^{3})

Apply log rule:

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

ln (x^{2} (x^{2} - 4)^{\frac{1}{2}}) - ln ((\frac{1}{x ^{2} + 2})^{3}) = ln \frac{x ^{2} ( x ^{2} - 4 ) ^{\frac{1}{2}}}{( \frac{1}{x ^{2} + 2} ) ^{3}}

= ln \frac{x ^{2} ( x ^{2} - 4 ) ^{\frac{1}{2}}}{( \frac{1}{x ^{2} + 2} ) ^{3}}

Simplify

\frac{x ^{2} ( x ^{2} - 4 ) ^{\frac{1}{2}}}{( \frac{1}{x ^{2} + 2} ) ^{3}} : x^{2} (x^{2} + 2)^{3} (x^{2} - 4)^{\frac{1}{2}}

= ln (x^{2} (x^{2} + 2)^{3} (x^{2} - 4)^{\frac{1}{2}})

ln (7 (n + 1))

8 lo g_{6} (u) + 7 lo g_{6} (v)

10 \cdot lo g_{10} (0.64 \cdot (π \cdot \frac{6.5}{0.0103})^{2})

- ln (3) - ln (2)

- ln (\frac{2}{5730})

What is 2ln(x)+1/2 ln(x^2-4)-3ln(1/(x^2+2)) ?
The solution to 2ln(x)+1/2 ln(x^2-4)-3ln(1/(x^2+2)) is ln(x^2(x^2+2)^3(x^2-4)^{1/2})