The logarithm is one of the final "basic" mathematical operations we will
observe. However while the initial concept of the logarithm may appear simple,
the study of logarithms has profound implications,
and was crucial in laying the foundation for many early concepts
in calculus. To put it simply, if 10 ^ x = y then log_10(y) = x.
Written neatly...
$$ \large \log_{10}(y) = x $$
We would say that the logarithm here is in "base 10". We might be able to easily
figure out that log_10(10) = 1, log_10(100) = 2, log_10(100000) = 5, and
log_10(1) = 0.
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The plotted graph of y = log_10(x).
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But it's still not apparent how we would calculate the value of log_10(6), for
example. Historically, this would be done using a small table of pre-determined
values and utilizing some clever properties of logarithms to approximate the
solution. These properties are:
$$ \large \log_b \left( n \cdot m \right) = \log_b \left( n \right) + \log_b \left( m \right) $$ $$ \large \log_b \left( \frac{n}{m} \right) = \log_b \left( n \right) - \log_b \left( m \right) $$ $$ \large \log_b \left( n^m \right) = m \cdot \log_b \left( n \right) $$
And here are a few common values:
$$ \large \log_{10}(2) \approx 0.301030 $$ $$ \large \log_{10}(3) \approx 0.477121 $$ $$ \large \log_{10}(5) \approx 0.698970 $$ $$ \large \log_{10}(7) \approx 0.845098 $$
Problem Statement
Using the above relationships and values provided above, approximate the base-10
logarithms of the given testcases.
Testcases will be chosen to ensure that an approximation is possible using only the provided values.
Please use the above approximation method, rather than using a built-in function
on a computer or calculator. In cases where the answers
would be different due to rounding error, the approximated answer will
be accepted rather than the exact one.
Input Data
First line will be Q, the quantity of testcases.
Q lines will then follow, each with a single integer.
Answer
Should consist of Q space-separated values, corresponding to the approximated
base-10 logarithm of each given integer.
Report the approximation to 6 decimal places.
Example
input data:
4
6
504
32400
5080320
answer:
0.778151 2.70243 4.510544 6.70589