Write as a sum and/or difference: a) `ln (^3sqrt(x^4+12)/((x+16)tsqrt(x-3)))` b) `log_5 (x^2(2+4x))^(3/2)/ sqrt(x^3-x)` Please explain the steps, this is so confusing. thank you!
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a) `ln ( ^3sqrt(x^4+12)/((x+16)sqrt(x-3)))`
`==> ln ((x^4+12)^(1/3) / ((x+16)(x-3)^(1/2)))`
`==> ln (x^4+12)^(1/3) - ln(x+16) - ln (x-3)^(1/2)`
`==> 1/3 ln (x^4+12) - ln (x^4+16) - 1/2 ln (x-3)`
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`b) log_5 ((x^2(2+4x))^(3/2) / sqrt(x^3-x))`
`==> log_5 (((x^2)^(3/2) (2+4x)^(3/2))/(x^3-x)^(1/2))`
`==> log_5 (x^2)^(3/2) + log_5 (2+4x)^(3/2) - log_5 (x^3-x)^(1/2)`
`==>log_5 x^3 +3/2 log_5 (2+4x) - log_5 (x(x^2-1))^(1/2)`
`==> 3log_5 x + 3/2 log_5 (2+4x) - log_5 x^(1/2) - log_5 (x^2-1)^(1/2)`
`==> 3log_5 x + 3/2log_5 (2+4x) - 1/2 log_5 x - 1/2 log_5(x^2-1)`
`==> 5/2 log_5 x + 3/2log_5 (2+4x) - 1/2 log_5 (x^2-1)`
`==> (5log_5 x + 3log_5 (2+4x) - log_5 (x^2-1))/2`
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