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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln ((3 x - 7)^{2}) = 2 ln (3 x - 7)

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

Apply log rule:

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

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

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

- (2 ln (3 x - 7) + 4 ln (5 x^{2} - 3 x)) : - 2 ln (3 x - 7) - 4 ln (5 x^{2} - 3 x)

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

2 ln (3) + 2 ln (2)

\frac{3}{2} \cdot ln^{3} (3)

- lo g_{10} (1.58 \cdot 1 0^{- 3})

\frac{3}{5} ln ∣ 4 - 1 ∣ + \frac{2}{5} ln ∣ 4 + 4 ∣

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

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