What is 2ln(x)+1/2 ln(x-1)-3ln(x+7) ?

The solution to 2ln(x)+1/2 ln(x-1)-3ln(x+7) is ln((x^2(x-1)^{1/2})/((x+7)^3))

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simplify 2ln(x)+1/2 ln(x-1)-3ln(x+7)

simplify

2 ln (x) + \frac{1}{2} ln (x - 1) - 3 ln (x + 7)

ln (\frac{x ^{2} ( x - 1 ) ^{\frac{1}{2}}}{( x + 7 ) ^{3}})

Solution steps

2 ln (x) + \frac{1}{2} ln (x - 1) - 3 ln (x + 7)

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 - 1) - 3 ln (x + 7)

Apply log rule:

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

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

= ln (x^{2}) + ln ((x - 1)^{\frac{1}{2}}) - 3 ln (x + 7)

Apply log rule:

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

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

= ln (x^{2} (x - 1)^{\frac{1}{2}}) - 3 ln (x + 7)

Apply log rule:

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

3 ln (x + 7) = ln ((x + 7)^{3})

= ln (x^{2} (x - 1)^{\frac{1}{2}}) - ln ((x + 7)^{3})

Apply log rule:

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

ln (x^{2} (x - 1)^{\frac{1}{2}}) - ln ((x + 7)^{3}) = ln (\frac{x ^{2} ( x - 1 ) ^{\frac{1}{2}}}{( x + 7 ) ^{3}})

= ln (\frac{x ^{2} ( x - 1 ) ^{\frac{1}{2}}}{( x + 7 ) ^{3}})

\frac{1}{24} ln cos (3 x^{2})

\frac{1}{3} lo g_{10} (5)

0.77 - \frac{8.134 \cdot 298}{1 \cdot 96500} \cdot ln (\frac{1}{4})

ln (\frac{0.01}{382})

5 - (- ln ∣ 8.7 - 3 ∣ + 2 ln ∣ 8.7 - 2 ∣)

What is 2ln(x)+1/2 ln(x-1)-3ln(x+7) ?
The solution to 2ln(x)+1/2 ln(x-1)-3ln(x+7) is ln((x^2(x-1)^{1/2})/((x+7)^3))