What is 3ln(x)-ln(3x)+ln(12x^2) ?

The solution to 3ln(x)-ln(3x)+ln(12x^2) is ln(4x^4)

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simplify 3ln(x)-ln(3x)+ln(12x^2)

simplify

3 ln (x) - ln (3 x) + ln (12 x^{2})

ln (4 x^{4})

Solution steps

3 ln (x) - ln (3 x) + ln (12 x^{2})

Apply log rule:

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

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

= ln (x^{3}) - ln (3 x) + ln (12 x^{2})

Apply log rule:

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

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

= ln (\frac{x ^{3}}{3 x}) + ln (12 x^{2})

Cancel the common factor:

x

= ln (\frac{x ^{2}}{3}) + ln (12 x^{2})

Apply log rule:

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

ln (\frac{x ^{2}}{3}) + ln (12 x^{2}) = ln (12 \cdot \frac{x ^{2}}{3} x^{2})

= ln (12 \cdot \frac{x ^{2}}{3} x^{2})

Simplify

\frac{x ^{2}}{3} \cdot 12 x^{2} : 4 x^{4}

= ln (4 x^{4})

ln (x^{2}) - 2 ln (y)

lo g_{10} (x y^{5})

lo g_{10} (x^{5} y^{4})

lo g_{5} (\frac{100}{3 x})

ln (\frac{( s ^{2} + c ^{2} ) ( s + a )}{( s + b ) ( s ^{2} - c ^{2} )})

What is 3ln(x)-ln(3x)+ln(12x^2) ?
The solution to 3ln(x)-ln(3x)+ln(12x^2) is ln(4x^4)