Rewrite this logarithm in a form with no logarithm of a product, quotient or power

Question

log {sqrt [(x^2 +16) / (x^2 +20)(x^3 -2)^2)] }

Answer 1

log√[(x²+16)/[(x²+20)(x³-2)²]]

= (1/2)log[(x²+16)/[(x²+20)(x³-2)²] <-- Exponent Rule

= (1/2)log[(x²+16)/(x²+20) * 1/(x³-2)²] <-- Write inside as a product of two fractions

= (1/2)log[(x²+16)/(x²+20)] + (1/2)log[1/(x³-2)²] <-- Use Sum Rule

= (1/2)log(x²+16) - (1/2)log(x²+20) + (1/2)log[(x³-2)^-2] <-- Use Difference Rule

= (1/2)log(x²+16) - (1/2)log(x²+20) - log(x³-2) <-- Exponent Rule

Hope this helped!