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

another rewrite is the exponential form

10^((x^2 +16)/(x^2+20)(x^3 -2))

although when you graph that on desmos, you get a slightly different graph than the graph of the log

both have a quadrant 1 hyperbola type shape though

the exponential has another branch not in the log graph

logs don't allow logs of negative values

while the exponential has no similar limitation for the domain

Answer 2

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!