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March 26, 2017

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We are given that log102<0.302. How many digits are there in the decimal representation of 5^500?

  • Maths - ,

    I am sure you meant:
    log10 2 < .302

    (you could have just said log 2 ≤ .302 , if the base is omitted it is assumed to be base 10)

    let x = 5^500
    = (10/2)^500
    log x = log (10/2)^500
    log x = 500(log (10/2)
    = 500(log10 - log2)
    = 500(1 - log2)
    = 500 - 500log2
    = 500 - 151 , = 349 from your given
    = 349.48 by using calculator for log2
    so your log x > 349

    what does that mean?

    if logx = 2 , then x = 10^2 ---100 --- 3 digits
    if logx = 3, then x = 10^3 -- 1000 --- 4 digits
    ...
    if logx = 349 , then x = 10^349 ----- 350 digits

    but log x > 349 , so it must contain 351 digits

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