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Logarithms
One of the most potent tools for investigating numbers is the common logarithm. Its uses will become apparent as you work more problems. For the moment, let's consider the following question:
Is it possible to get the common log values of all the single digit integers just by knowing the log values of a few of them?
It can be shown using the rules of logs that nearly all of them can be generated from
and using only the four basic operations
however there is one log value that we can't generate so easily using these two. Namely,
.
Green Question: How could I find a reasonably good estimate of the value of just using the rules of logas and those two given above?
i.) In your thoughts, consider whether your estimation is an overestimate or an underestimate.
ii.) Determine more than one estimate.
iii.) Finally, only after you've got the estimates you want, use the log base 10 key on your calculator to determine which of your estimates turned out to be best.
Solution:
Blue Question: What is the remainder when 10 divides 21000 ?
A complete solution to this problem is posted below.
Orange Question: How many digits are in the number 21000 ?
STEP 2000, Math I, #1: (no calculator) To nine decimal places,
and .
i.) Calculate
and to three decimal places. By taking logs, or otherwise,
show that
.
Hence write down the first digit of
ii.) Find the first digit of each of the following numbers:
; ; and .
A complete solution to this problem is posted below.
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