Write as the sum and/or difference of logarithms. Express powers as factors.
-$\ln \left( \frac { ( x + 5 ) ( x - 2 ) } { ( x - 8 ) ^ { 4 } } \right) ^ { 4 / 3 } , \quad x 2$
A) $4 \ln ( x + 5 ) - 3 \ln ( x - 2 ) - \frac { 16 } { 3 } \ln ( x - 8 )$
B) $\frac { 4 } { 3 } \ln \left( x ^ { 2 } + 7 x - 10 \right) - \frac { 16 } { 3 } \ln ( x - 8 )$
C) $\ln ( x + 5 ) + \ln ( x - 2 ) + \ln 4 - 16 \ln ( x - 8 ) - \ln 3$
D) $\frac { 4 } { 3 } \ln ( x + 5 ) + \frac { 4 } { 3 } \ln ( x - 2 ) - \frac { 16 } { 3 } \ln ( x - 8 )$

Question

Write as the sum and/or difference of logarithms. Express powers as factors.
-$\ln \left( \frac { ( x + 5 ) ( x - 2 ) } { ( x - 8 ) ^ { 4 } } \right) ^ { 4 / 3 } , \quad x > 2$
A) $4 \ln ( x + 5 ) - 3 \ln ( x - 2 ) - \frac { 16 } { 3 } \ln ( x - 8 )$
B) $\frac { 4 } { 3 } \ln \left( x ^ { 2 } + 7 x - 10 \right) - \frac { 16 } { 3 } \ln ( x - 8 )$
C) $\ln ( x + 5 ) + \ln ( x - 2 ) + \ln 4 - 16 \ln ( x - 8 ) - \ln 3$
D) $\frac { 4 } { 3 } \ln ( x + 5 ) + \frac { 4 } { 3 } \ln ( x - 2 ) - \frac { 16 } { 3 } \ln ( x - 8 )$

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be rewritten as the sum and difference of individual logarithms, with the exponent $4/3$ applied to each term and the division inside the logarithm turned into subtraction.

Write as the Sum And/or Difference of Logarithms ln⁡((x+5)(x−2)(x−8)4)4/3,x>2\ln \left( \frac { ( x + 5 ) ( x - 2 ) } { ( x - 8 ) ^ { 4 } } \right) ^ { 4 / 3 } , \quad x > 2ln((x−8)4(x+5)(x−2)​)4/3,x>2

Write as the Sum And/or Difference of Logarithms $\ln \left( \frac { ( x + 5 ) ( x - 2 ) } { ( x - 8 ) ^ { 4 } } \right) ^ { 4 / 3 } , \quad x > 2$