My question is how did the last step come about I dont under

My question is how did the last step come about? I don\'t understand where the 3/2 came from and how the x went from being on both the top and the bottom to just the bottom. Can some one explain to me. What log rule?

Solution

1/2 log _a x + 4 log _a y - 2 log _a x

applying log properties

n log m = log m^n

log m + log n = log (m*n )

log m - log n = log ( m/n)

therefore ,

1/2 log _a x + 4 log _a y - 2 log _a x = log _a x^1/2 + log _a y^4 - log _a x^2

now , log _a x^1/2 + log _a y^4 - log _a x^2 = log _a ( x^1/2 * y^4 / x^2 )

applying exponent property

a^m / a^n = 1 / a ^( n-m)

subtracting exponent of x at the top from the exponent of x at the bottom

log _a ( x^1/2 * y^4 / x^2 ) = log _a ( y^4 / x^(2-1/2) ) = log _a ( y^4 / x^3/2 )