A Plot of Decibels vs N over four decades of range:
Logarithms are related to decibels, which
are used to define power ratios over a very wide range. dB = 20 x log(N)
for the base ten. In engineering N represents a voltage or current
level. Therefore, if N = 10, log(10) =1 and we have a 20 dB or
100 times increase in power level for a ten times increase in signal
level. Similarly, 20 x log(2) = 6.02 dB for twice the voltage level.
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1. Question: How can this equation be solved?
3logx + 5logx + 6logx = N.
Answer: The solution requires a method that
is sort of the inverse of what is described on this web site.
Nevertheless, a similar approach can be used:
The first step is to reduce the equation arithmetically:
3logx + 5logx + 6logx = 14 logx = 64
logx = 62/14 = 32/7 = 4.57 = 4 + 0.57 (note that
x = log10,000).
Note that the inverse log of 4 is N = 10,000,
which is obtained by taking 1 (log1 = 0) and moving the decimal point four
places to the right. This value is the first rough estimate of N, but we
want a much closer estimate. For base 10, we will first find the log of
0.57 and then move the decimal point four places to the right. As
stated earlier, our number to remember is
log 2 = 0.301,
so log 4 = log 2exp2 = 2 x log 2 = 0.602,
which is fairly close to 0.57. Therefore, the first
estimate must be multiplied by
x ~ 4,
and our second rough estimate becomes N = 40,000.
We know that this estimate is slightly high, since it 0.602 is higher than
Using linear interpolation,
0.602 - 0.57 = 0.032,
which is an error of 0.032/0.602 = 5.3%, and we must
compensate by 1/1.053.
Dividing x by this amount,
4/1.053 = 3.79
Moving the decimal point four places to the right,
our estimate of N becomes:
N ~ 37,900.
The actual value is:
N = 37,153
and the error in the estimate is +2.01%.
As you can see, this method is not perfect, but it
is fairly accurate. The accuracy of the estimate can be increased, which
is the approach that I had originally prepared but it appeared to be too
complex for most readers to prefer to absorb on their first short look.