Step-by-step explanation:
Begin by separating the division. Division in the argument of a logarithm function is equivalent to subtracting the log of the numerator and denominator:
[tex]ln( \frac{ {x}^{9} \sqrt{ {x}^{2} + 7} }{ {(x + 7)}^{7} } ) = ln( {x}^{9} \sqrt{ {x}^{2} + 7 } ) - ln(( {x + 7})^{7} )[/tex]
Multiplication under a logarithm means addition:
[tex] ln( {x}^{9}) + ln( \sqrt{ {x}^{2} + 7 }) - ln(( {x + 7})^{7} )[/tex]
Log rules say that exponents can be dragged to the front as coefficients:
[tex]9ln(x) + \frac{1}{2} ln( {x}^{2} + 7) - 7ln(x + 7)[/tex]
