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Math Help - logarithms question

  1. #1
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    logarithms question

    can someone please explain how to solve these questions?

    The number of mold spores in a petri dish increases by a factor of 10 every week. If there are initially 40 spores in the dish, how long with it take for there to be 2000 spores?

    and

    The astronomer johannes kepler determined that the time, D, in days, for a planet to revolve around the sun is related to the planet's average distance from the sun, k, in millions of kilometres. This relation is defined by the equation log D = (3/2)logk - 0.7. Estimate the period of revolution of uranus about the sun, given its distance from the sun. Uranus is 2854 million kilometres away from the sun
    Last edited by mzto; May 9th 2009 at 05:34 PM.
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  2. #2
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    Hello mzto
    Quote Originally Posted by mzto View Post
    can someone please explain how to solve these questions?

    The number of mold spores in a petri dish increases by a factor of 10 every week. If there are initially 40 spores in the dish, how long with it take for there to be 2000 spores?
    At time t = 0, there are 40 spores.

    After 1 week, t=1, there are 40\times 10 spores

    After 2 weeks, t=2, there are 40\times 10\times 10 = 40\times 10^2 spores

    ... and so on.

    After t weeks, there are 40\times 10^t spores

    So, if there are 2000 spores at time t, 40\times 10^t = 2000

    \Rightarrow 10^t = \frac{2000}{40}= 50

    \Rightarrow t=\log(50), because the log (to base 10) of a number is the power to which you must raise 10 to get the number. (Do you understand this sentence? It's vital that you do, because you'll need it again in the second question - see below!)

    \Rightarrow t =1.699 weeks, or 11 days 21.4 hours.

    The astronomer johannes kepler determined that the time, D, in days, for a planet to revolve around the sun is related to the planet's average distance from the sun, k, in millions of kilometres. This relation is defined by the equation log D = (3/2)logk - 0.7. Estimate the period of revolution of uranus about the sun, given its distance from the sun. Uranus is 2854 million kilometres away from the sun
    We need to find D when k = 2854.

    So \log D = \tfrac32\log(2854) - 0.7

    = 4.48318...

    So 4.48318 is the power to which we must raise 10 in order to get D (see question 1). In other words:

    D = 10^{4.48318} = 30421 days

    (Note that you can probably do this calculation on your calculator in two ways: one is to enter 4.48318 and then press INV and LOG, and the other is to use the x^y button, entering 10 for x and 4.48318 for y.)

    Grandad
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