What is 5ln(x+3)+3ln(2x)-4ln(x-1) ?

The solution to 5ln(x+3)+3ln(2x)-4ln(x-1) is ln((8x^3(x+3)^5)/((x-1)^4))

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simplify 5ln(x+3)+3ln(2x)-4ln(x-1)

simplify

5 ln (x + 3) + 3 ln (2 x) - 4 ln (x - 1)

ln (\frac{8 x ^{3} ( x + 3 ) ^{5}}{( x - 1 ) ^{4}})

Solution steps

5 ln (x + 3) + 3 ln (2 x) - 4 ln (x - 1)

Apply log rule:

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

5 ln (x + 3) = ln ((x + 3)^{5})

= ln ((x + 3)^{5}) + 3 ln (2 x) - 4 ln (x - 1)

Apply log rule:

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

3 ln (2 x) = ln ((2 x)^{3})

= ln ((x + 3)^{5}) + ln ((2 x)^{3}) - 4 ln (x - 1)

Apply log rule:

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

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

= ln ((x + 3)^{5} (2 x)^{3}) - 4 ln (x - 1)

Simplify

(x + 3)^{5} (2 x)^{3} : 8 x^{3} (x + 3)^{5}

= ln (8 x^{3} (x + 3)^{5}) - 4 ln (x - 1)

Apply log rule:

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

4 ln (x - 1) = ln ((x - 1)^{4})

= ln (8 x^{3} (x + 3)^{5}) - ln ((x - 1)^{4})

Apply log rule:

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

ln (8 x^{3} (x + 3)^{5}) - ln ((x - 1)^{4}) = ln (\frac{8 x ^{3} ( x + 3 ) ^{5}}{( x - 1 ) ^{4}})

= ln (\frac{8 x ^{3} ( x + 3 ) ^{5}}{( x - 1 ) ^{4}})

ln (7) + 3 ln (2)

ln (7) + 3 ln (4)

ln (\frac{13000}{10815})

ln (8) + 4 ln (3)

- ln (6) + ln (2)

What is 5ln(x+3)+3ln(2x)-4ln(x-1) ?
The solution to 5ln(x+3)+3ln(2x)-4ln(x-1) is ln((8x^3(x+3)^5)/((x-1)^4))