What is ln((x^4)/((2x-1)^3(7x-5)^8)) ?

The solution to ln((x^4)/((2x-1)^3(7x-5)^8)) is 4ln(x)-3ln(2x-1)-8ln(7x-5)

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simplify ln((x^4)/((2x-1)^3(7x-5)^8))

simplify

ln (\frac{x ^{4}}{( 2 x - 1 ) ^{3} ( 7 x - 5 ) ^{8}})

4 ln (x) - 3 ln (2 x - 1) - 8 ln (7 x - 5)

Solution steps

ln (\frac{x ^{4}}{( 2 x - 1 ) ^{3} ( 7 x - 5 ) ^{8}})

Apply log rule:

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

ln (\frac{x ^{4}}{( 2 x - 1 ) ^{3} ( 7 x - 5 ) ^{8}}) = ln (x^{4}) - ln ((2 x - 1)^{3} (7 x - 5)^{8})

= ln (x^{4}) - ln ((2 x - 1)^{3} (7 x - 5)^{8})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= 4 ln (x) - ln ((2 x - 1)^{3} (7 x - 5)^{8})

Apply log rule:

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

ln ((2 x - 1)^{3} (7 x - 5)^{8}) = ln ((2 x - 1)^{3}) + ln ((7 x - 5)^{8})

= 4 ln (x) - (ln ((2 x - 1)^{3}) + ln ((7 x - 5)^{8}))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= 4 ln (x) - (3 ln (2 x - 1) + ln ((7 x - 5)^{8}))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((7 x - 5)^{8}) = 8 ln (7 x - 5)

= 4 ln (x) - (3 ln (2 x - 1) + 8 ln (7 x - 5))

- (3 ln (2 x - 1) + 8 ln (7 x - 5)) : - 3 ln (2 x - 1) - 8 ln (7 x - 5)

= 4 ln (x) - 3 ln (2 x - 1) - 8 ln (7 x - 5)

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

lo g_{10} (\frac{y ^{2}}{2 z})

lo g_{10} (18) - lo g_{10} (6)

lo g_{3} (2) + lo g_{3} (13)

2 lo g_{5} (x) - lo g_{5} (y) - \frac{1}{2} lo g_{5} (z)

What is ln((x^4)/((2x-1)^3(7x-5)^8)) ?
The solution to ln((x^4)/((2x-1)^3(7x-5)^8)) is 4ln(x)-3ln(2x-1)-8ln(7x-5)