Question 244998
you are given:


log(a,x) = 4
log(a,y) = 3
log(a,z) = 2


you want to find:


{{{log(a,(((root(5,y^3*x^6*z^4)))/(root(5,(z^6*x^2)))))}}}


since {{{root(n,a)/root(n,b) = root(n,(a/b))}}}, your equation becomes:


{{{log(a,((root(5,(y^3*x^6*z^4)/(z^6*x^2)))))}}}


since {{{root(5,a) = a^(1/5)}}}, your equation becomes:


{{{log(a,(((y^3*x^6*z^4)/(z^6*x^2))^(1/5))))}}}


since log(a^b) = b*(log(a), your equation becomes:


{{{(1/5) * log(a,(((y^3*x^6*z^4)/(z^6*x^2)))))}}}


since log(a/b) = log(a)/log(b), your equation becomes:


{{{(1/5) * (log(a,(y^3*x^6*z^4)) - log(a,(z^6*x^2)))}}}


since log(a*b) = log(a) + log(b), your equation becomes:


{{{(1/5) * (log(a,(y^3)) + log(a,(x^6)) + log(a,(z^4)) - log(a,(z^6)) - log(a,(x^2)))}}}


since log(a^b) = b*log(a), your equation becomes:


{{{(1/5) * (3*log(a,(y)) + 6*log(a,(x)) + 4*log(a,(z)) - 6*log(a,(z)) - 2*log(a,(x)))}}}


from here on it's just a straight substitution since you are given:


log(a,x) = 4
log(a,y) = 3
log(a,z) = 2


your equation of:


{{{(1/5) * (3*log(a,(y)) + 6*log(a,(x)) + 4*log(a,(z)) - 6*log(a,(z)) - 2*log(a,(x)))}}}


becomes:


{{{(1/5) * (3*3 + 6*4 + 4*2 - 6*2 - 2*4)}}} which becomes:


{{{(1/5) * (9+24+8-12-8)}}} which becomes:


{{{(1/5) * (21)}}} which becomes:


{{{21/5}}}