Logarithm Help: Logs Given, Find Blank.?

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.