Question

Use the properties of logarithms to write the following as sum and/or difference of logarithms:

log√[(x^2 -1)/(x^3)((y^2+1)^5)]

Answer 1

ok i guess we can straighten out this mess

Answer 2

is the square root over the whole input, or just in the numerator

Answer 3

It's over the whole thing

Answer 4

the the first step is to get rid of the square root by putting a \(\frac{1}{2}\) out front

Answer 5

\[\frac{1}{2}\log\left(\frac{(x^2-1)}{x^3(y^2+1)^5}\right)\]

Answer 6

the turn the division in to a subtraction
\[\frac{1}{2}\log(x^2-1)-\log(x^3(y+1)^5)\]

Answer 7

then turn the product in to a sum
\[\frac{1}{2}\log(x^2-1)-\left(\log(x^3)+\log((y+1)^5)\right)\]

Answer 8

factor and pull out the exponents as multipliers
[\frac{1}{2}\left(\log((x+1)(x-1))-3\log(x)-5\log(x+1)\right)\]

Answer 9

\[\frac{1}{2}\left(\log((x+1)(x-1))-3\log(x)-5\log(x+1)\right)\]

Answer 10

one last step
\[\frac{1}{2}\left(\log((x+1)+\log(x-1)-3\log(x)-5\log(x+1)\right)\]