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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

Apply log rule:

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

5 ln (x - 1) = ln ((x - 1)^{5})

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

Apply log rule:

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

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

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

Simplify

\frac{x ^{3} 3 x + 1}{3 x + 1} (x - 1)^{5} : \frac{x ^{3} ( x - 1 ) ^{5} 3 x + 1}{3 x + 1}

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

Apply log rule:

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

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

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

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

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

lo g_{10} (\frac{x}{x + 1})

3 lo g_{10} (a) - lo g_{10} (q)

ln (x^{2} - 9) + 2 ln (\frac{1}{x + 3}) + 4 ln (x)

lo g_{10} (w^{6} u \cdot v)

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