What is 3(ln(x)-2ln(x^2+1))+2ln(5) ?

The solution to 3(ln(x)-2ln(x^2+1))+2ln(5) is ln((25x^3)/((x^2+1)^6))

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simplify 3(ln(x)-2ln(x^2+1))+2ln(5)

simplify

3 (ln (x) - 2 ln (x^{2} + 1)) + 2 ln (5)

ln (\frac{25 x ^{3}}{( x ^{2} + 1 ) ^{6}})

Solution steps

3 (ln (x) - 2 ln (x^{2} + 1)) + 2 ln (5)

Apply log rule:

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

2 ln (x^{2} + 1) = ln ((x^{2} + 1)^{2})

= 3 (ln (x) - ln ((x^{2} + 1)^{2})) + 2 ln (5)

Apply log rule:

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

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

= 3 ln (\frac{x}{( x ^{2} + 1 ) ^{2}}) + 2 ln (5)

Apply log rule:

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

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

= ln (\frac{x}{( x ^{2} + 1 ) ^{2}})^{3} + 2 ln (5)

Apply log rule:

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

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

= ln (\frac{x}{( x ^{2} + 1 ) ^{2}})^{3} + ln (5^{2})

Apply log rule:

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

ln (\frac{x}{( x ^{2} + 1 ) ^{2}})^{3} + ln (5^{2}) = ln 5^{2} (\frac{x}{( x ^{2} + 1 ) ^{2}})^{3}

= ln 5^{2} (\frac{x}{( x ^{2} + 1 ) ^{2}})^{3}

Simplify

(\frac{x}{( x ^{2} + 1 ) ^{2}})^{3} \cdot 5^{2} : \frac{25 x ^{3}}{( x ^{2} + 1 ) ^{6}}

= ln (\frac{25 x ^{3}}{( x ^{2} + 1 ) ^{6}})

lo g_{8} (4) + 2 lo g_{8} (x)

ln (\frac{23}{7})

lo g_{10} (27) + lo g_{10} (\frac{1}{9})

ln (\frac{5}{9 x ^{9} y})

lo g_{3} (7) - lo g_{3} (8)

What is 3(ln(x)-2ln(x^2+1))+2ln(5) ?
The solution to 3(ln(x)-2ln(x^2+1))+2ln(5) is ln((25x^3)/((x^2+1)^6))