I've been stuck on these four all night, any help would be appreciated!
1) Given: log_4^(y)=3.65, log_4^(t)=4.45, log_4^(z)=-13.08
Find: log_4(y^10t^2/z^7)
2) Given: log_7^(y)=-4.67, log_7^(z)=4.43, log_7^(x)= 19.06
Find: -18log_7(z^11/y^9x^8)
3) log_6^(y)=1.62, log_6^(x)=14.42, log_6^(t)=9.34
Find: log_6( root (2) sqrt (x^11)/y^6t^5)
4) Given: log_4^(z)=5.76, log_4^(y)=9.82, log_4^(x)=9
Find: log_4( root (4) sqrt (y^3/z^7x^5))
greenhart
We make note of the three main properties of logs.
A Log (ab)= log a + log b
B log (a^z)= zlog a
C log (a/b) = log a - log b.
I will cite these as we go
Ready?
1) find: log_4(y^10t^2/z^7)
notice the z^7 in the denominator? Let's get rid of that by using property C
log_4(y^10t^2/z^7) = log_4(y^10t^2) - log_4 (z^7)
Now we have a product y^10 times t^2, we use property A for that one!
log_4(y^10t^2) - log_4 (z^7) = log_4(y^10) + log_4(t^2) - log_4 (z^7)
Now let's just get rid of those nasty exponents using property B.
log_4(y^10) + log_4(t^2) - log_4 (z^7) = 10log_4(y) + 2 log_4(t) - 7log_4(z)
All right: now all we need to do is to use the given information and plug in!
we get 10log_4(y) + 2 log_4(t) - 7log_4(z)= 10(3.65) + 2(4.45) - 7(-13.08)
google says this is 136.96
2) Find: -18log_7(z^11/y^9x^8)
We do the same thing. First use property C to get the stuff out of denominator (ignore the -18 on the outside for now. You can just multiply that at the very end.)
log_7(z^11/y^9x^8) =
log_7(z^11) - log_7 (y^9) - log (x^8).
We can actually go right to property B and get rid of exponents
log_7(z^11) - log_7 (y^9) - log (x^8)= 11log_7(z) - 9log_7(y) - 8log(x)
Plug in given info to get 11log_7(z) - 9log_7(y) - 8log_7(x)= 11(4.43)-9(-4.67)-8(19.06)
Google claims this is -61.72. But them multiply by the - 18 to get 1110.96
3 and 4 are the same type of deal. I'm going to go through them faster without citing the properties.
3.
log_6( root (2) sqrt (x^11)/y^6t^5) = log_6 root 2 + log_6 x^11 - log_6 y^6 - log_6 t^5) =
log_6 root 2 + 11 log_6 x - 6 log_6 y - 4 log_6 t =
log_6 root 2 + 11 (14.42) - 6(1.62) - 4 (9.34) = 11.92 approximately.
4. log_4 root (4) sqrt (y^3/z^7x^5)) =
log_4root (4) + log_4(y^3/z^7x^5)^1/2=
log_4 root (4) + 1/2log_4(y^3/z^7x^5) (property B) =
log_4 root (4) + 1/2[log_4(y^3)-log_4(z^7)-log_4(x^5)] =
log_4 root (4) + 1/2[3log_4(y) - 7 log_4(z) - 5 log_4(x)] =
log_4 root (4) + 1/2 [3(9.82)-7(5.76)-5(9)= -27.43
Now, I may have done some errors since many steps. So I advise you to check them through with paper (instead of computer and virtual calculator), but hopefully you see the process better.