calculus -- PLEASE HELP!

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.

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  1. 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

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  2. 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 :)

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