Question 1122163
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The "^" symbol is used to represent exponents -- not to indicate the base of a logarithm.  Your question is<br>
Given that {{{log(8,(p+2))+log(8,(q))=r-1/3}}} and {{{log(2,(p-2))-log(2,(q))=2r+1}}} then show that {{{p^2 = 4+32^r}}}<br>
It looks ugly (especially the way you showed it!), but everything falls in place nicely using basic rules of logarithms.  Specifically, we need to use<br>
log(a)+log(b) = log(ab)
log(a)-log(b) = log(a/b)
log base 2 of x = 3*log base 8 of x, since 8 = 2^3<br>
(1) {{{log(8,(p+2))+log(8,(q))=r-1/3}}}  Given<br>
(2) {{{log(2,(p-2))-log(2,(q))=2r+1}}}  Given<br>
(3) {{{log(8,(q(p+2)))=r-1/3}}}  from (1)<br>
(4) {{{log(2,(q(p+2))) = 3r-1}}}  from (3)<br>
(5) {{{log(2,((p-2)/q)) = 2r+1}}}  from (2)<br>
(6) {{{log(2,(p^2-4)) = 5r}}}  from (4) and (5)<br>
(7) {{{p^2-4 = 2^(5r) = 32^r}}}  (definition of log base 2)
(8) {{{p^2 = 4+32^r}}}