Question

Use properties of logarithms to expand the logarithmic expression as much as possible.

help please!

Answer 1

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:

`ln( \frac{ {x}^(9) \sqrt{ {x}^(2) + 7} }{ {(x + 7)}^(7) } ) = ln( {x}^(9) \sqrt{ {x}^(2) + 7 } ) - ln(( {x + 7})^(7) )`

Multiplication under a logarithm means addition:

`ln( {x}^(9)) + ln( \sqrt{ {x}^(2) + 7 }) - ln(( {x + 7})^(7) )`

Log rules say that exponents can be dragged to the front as coefficients:

`9ln(x) + (1)/(2) ln( {x}^(2) + 7) - 7ln(x + 7)`