log trouble

I'm having trouble with logs where different bases are used, can someone please explain how to do them. Here's and example.

a. Show that Log43 = log2root3

I can do part a.... but i can't do part b.

b. Hence or otherwise solve the simultaneous equations

2log2Y = log43 + log2X
3^y = 9^x
given that x and y are positive.

I appreciate any help i can get.

Reply 1

Jonny W

It follows from 3^y = 9^x that y = 2x, and from

2 log[2](y) = log[4](3) + log[2](x)

that

2 log[2](y) = log[2](sqrt(3)) + log[2](x) . . . . . (*)

--

Putting y = 2x into (*) gives

2 log[2](2) + 2 log[2](x) = log[2](sqrt(3)) + log[2](x)

log[2](x)
= log[2](sqrt(3)) - 2 log[2](2)
= log[2](sqrt(3)/4)

x = sqrt(3)/4
y = sqrt(3)/2