1. To five decimals it is a fact that log(2)=.30103 Compute the following to three decimal places usuing ONLY this fact and the logarithmic rules and the definition of a logarithm.

a. log100 (2) --base 100
b. log1000 (square root of 2) -- base 1000

2. Suppose you know that logb(C) = D. If logb(E)=D^2, find E in terms of some (or all) of the terms b, C, and D.

base^log base ( x )= x

log100 (2) = log 10(2) * log100(10)
but
100^log 100(10) = 10
so log100 (10) = .5
because
100^.5 = 10
so
log 100(2) = .30103*.5
=0.151 <------ answer
check
100^.151 = ?
2 sure enough
others similar - will continue but have to go now

b.

log 1000 (2^.5) = .5 log 1000 (2)
log 1000(2) = log 10(2) log 1000(10)

log 1000(10) = 1/3
because 1000^(1/3) = 10

so
.5 log1000(2) = .5(.30103)/3
=.0502 <----- answer

check
1000^.0502 =?
1.414 sure enough sqrt 2

2. Well, it does not matter what base so I will use ten and just write log
D = log C
D^2 = log C * log C
but D^2 = log E
so
log E = log C * log C
so
10^log E = E = 10^(log C*log C)
so
E = [10^log C]^log C
but 10^log C = C
so
E = C^log C
but log C = D
so
E = C^D
remarkable :)

Sure! Let's solve each problem step by step:

1. a. log100 (2) -- base 100:
We can use the logarithmic rule that states: log base A of B is equal to log base C of B divided by log base C of A.
To solve this, we can express log100 (2) in terms of logarithms with a base of 10 (since we know the value for log(2)):
log100 (2) = log(2) / log(100)

Now, we substitute the known values:
log100 (2) = 0.30103 / log(100)

Next, we apply another logarithmic rule: log(base A) of B can be written as log(base 10) of B divided by log(base 10) of A.
So, we can express log(100) as log(base 10) of 100 divided by log(base 10) of 10:
log100 (2) = 0.30103 / (log(100) / log(10))

Simplifying further:
log100 (2) = 0.30103 / (2 / 1) [log(100) equals 2 and log(10) equals 1]

Finally:
log100 (2) ≈ 0.30103 / 2 ≈ 0.150515

Therefore, log100 (2) to three decimal places is approximately 0.151.

1. b. log1000 (square root of 2) -- base 1000:
Using similar logarithmic rules as before, we can proceed:

log1000 (square root of 2) = (log(square root of 2)) / log(1000)
Since we know log(2) = 0.30103, we can substitute:
log1000 (square root of 2) = (log(square root of 2)) / log(1000)
log1000 (square root of 2) = (0.30103 / 2) ≈ 0.150515

Therefore, log1000 (square root of 2) to three decimal places is approximately 0.151.

2. If logb(C) = D, and logb(E) = D^2, we need to find E in terms of b, C, and D.
Given that logb(C) = D, we can rewrite it as b^D = C.

Now, we are given logb(E) = D^2.
Using the same logic, we can rewrite it as b^(D^2) = E.

Therefore, E can be expressed as b raised to the power of D squared, or E = b^(D^2).

So, in terms of the given variables, E = b^(D^2).