What is ln(((x+5)^2)/((x-9)^2(x^2-15)^5)) ?

The solution to ln(((x+5)^2)/((x-9)^2(x^2-15)^5)) is 2ln(x+5)-2ln(x-9)-5ln(x^2-15)

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simplify ln(((x+5)^2)/((x-9)^2(x^2-15)^5))

simplify

ln (\frac{( x + 5 ) ^{2}}{( x - 9 ) ^{2} ( x ^{2} - 15 ) ^{5}})

2 ln (x + 5) - 2 ln (x - 9) - 5 ln (x^{2} - 15)

Solution steps

ln (\frac{( x + 5 ) ^{2}}{( x - 9 ) ^{2} ( x ^{2} - 15 ) ^{5}})

Apply log rule:

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

ln (\frac{( x + 5 ) ^{2}}{( x - 9 ) ^{2} ( x ^{2} - 15 ) ^{5}}) = ln ((x + 5)^{2}) - ln ((x - 9)^{2} (x^{2} - 15)^{5})

= ln ((x + 5)^{2}) - ln ((x - 9)^{2} (x^{2} - 15)^{5})

Apply log rule:

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

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

= 2 ln (x + 5) - ln ((x - 9)^{2} (x^{2} - 15)^{5})

Apply log rule:

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

ln ((x - 9)^{2} (x^{2} - 15)^{5}) = ln ((x - 9)^{2}) + ln ((x^{2} - 15)^{5})

= 2 ln (x + 5) - (ln ((x - 9)^{2}) + ln ((x^{2} - 15)^{5}))

Apply log rule:

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

ln ((x - 9)^{2}) = 2 ln (x - 9)

= 2 ln (x + 5) - (2 ln (x - 9) + ln ((x^{2} - 15)^{5}))

Apply log rule:

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

ln ((x^{2} - 15)^{5}) = 5 ln (x^{2} - 15)

= 2 ln (x + 5) - (2 ln (x - 9) + 5 ln (x^{2} - 15))

- (2 ln (x - 9) + 5 ln (x^{2} - 15)) : - 2 ln (x - 9) - 5 ln (x^{2} - 15)

= 2 ln (x + 5) - 2 ln (x - 9) - 5 ln (x^{2} - 15)

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What is ln(((x+5)^2)/((x-9)^2(x^2-15)^5)) ?
The solution to ln(((x+5)^2)/((x-9)^2(x^2-15)^5)) is 2ln(x+5)-2ln(x-9)-5ln(x^2-15)