What is ln(((t+5)^3)/(t^4(5-1)^2)) ?

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

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simplify ln(((t+5)^3)/(t^4(5-1)^2))

simplify

ln (\frac{( t + 5 ) ^{3}}{t ^{4} ( 5 - 1 ) ^{2}})

3 ln (t + 5) - 4 ln (t) - 4 ln (2)

Solution steps

ln (\frac{( t + 5 ) ^{3}}{t ^{4} ( 5 - 1 ) ^{2}})

Apply log rule:

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

ln (\frac{( t + 5 ) ^{3}}{t ^{4} ( 5 - 1 ) ^{2}}) = ln ((t + 5)^{3}) - ln (t^{4} (5 - 1)^{2})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln ((5 - 1)^{2}) = 2 ln (5 - 1)

= 3 ln (t + 5) - (4 ln (t) + 2 ln (5 - 1))

2 ln (5 - 1) = 4 ln (2)

= 3 ln (t + 5) - (4 ln (t) + 4 ln (2))

- (4 ln (t) + 4 ln (2)) : - 4 ln (t) - 4 ln (2)

= 3 ln (t + 5) - 4 ln (t) - 4 ln (2)

ln (x^{5}) - ln (x)

- 4 ln (x) + ln (5 y) - ln (4 z)

\frac{1}{3} lo g_{10} (a) - \frac{1}{2} (lo g_{10} (b) - lo g_{10} (c))

lo g_{2} (6) - lo g_{2} (15) - lo g_{2} (20)

lo g_{10} (4 x)

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