Significant Figures in Logarithms and Anti-logs. In this video, I will be showing you how to
properly round calculated answers after you use logarithms or anti-logs. I am assuming that you
already know what significant figures are and why they're important, and that you can look
at a number and determine how many significant figures are in it. If you do not already know
those two things, I recommend watching my first two videos in this playlist before you proceed.
The first thing you need to know is that there is a different significant figures rule
for each mathematical operation, so there is a rule for logs and anti-logs,
there's a rule for multiplication and division, and there is a rule for addition and subtraction.
Right now, I am only talking about logarithms and anti-logs. But, our one and only rule is:
The number of significant figures after the decimal
in a calculated log value equals the number of significant figures in the original number.
Let's look at some examples involving logarithms. So in this column,
I have a bunch of numbers, and in this column I have the logs of those numbers. In other words,
the log of 1.8 times 10 to the minus 9th is equal to negative 8.744727.
Our goal is to properly round these numbers. These are the numbers that
a calculator would give and calculators do not care about significant figures.
They give as many digits as they possibly can. So we need to apply our rule to round these numbers.
One thing I want to point out: These first three numbers,
they all have the same coefficient. They are all 1.8 times 10 to some power, and the logarithms
of those values all have the same digits after the decimal place. They are all something.744727
This means that the coefficient is controlling digits after the decimal and the exponent is
controlling digits before the decimal. That is the key to our role. So this calculated logarithm
has as many significant figures after the decimal place as this original number has in its entirety.
So 1.8 has two significant figures. The log of that value has two significant figures after the
decimal. We are always considering anything before the decimal point as significant. So overall, this
has three significant figures. We need to round this number to negative 8.74.
The same applies for these next two: They both have two significant figures in the coefficient,
so everything is significant up through two digits after the decimal place.
But notice what just happened here. Two of these numbers have three significant
figures overall the third one has four significant figures overall.
The constant here is two significant figures in the coefficient of the original number
means two significant figures after the decimal place. Let's look at these last two examples:
We have a different coefficient, 6.21, so we have different digits after the decimal place.
But the same concept applies: The log of 6.21 times 10 to the 8th equals 8.793092
6.21 has three significant figures,
so this log has three significant figures after the decimal point. We need to round this to 8.793
In our final example, 6.21 times 10 to the 102 power, same concept: Three significant figures.
Everything is significant up through three digits after the decimal place,
so this needs to be rounded to 102.793
Now, let's work on rounding some anti-logs. So logarithms and anti-logarithms are reverse of
the same processes. So I need to take that rule I gave you earlier and reverse it.
But, same setup as before. I have values for x and I have calculated anti-logs of those values.
So the way that you calculate an anti-log when you're talking about log base 10
is to take 10 to the power of that number. So for example, our first number is negative 4.74.
The anti-log would be 10 to the power of negative 4.74. The answer that a calculator would give
you is 1.81970 times 10 to the minus fifth. But let's apply our previous role in reverse. Think
of these as what we were previously calling the calculated log values. What matters here is how
many significant figures are after the decimal point. We do not care how many significant
figures are before the decimal point. Before the decimal point is controlling this exponent.
After the decimal point is controlling the coefficient. So there are two significant figures
after the decimal point here, so the anti-log of this number needs two significant figures overall.
I need to round this to 1.8 times 10 to the minus fifth. Our second example, negative 10.74,
still has two significant figures after the decimal. The first number had three
significant figures overall second number had four significant figures overall but they both have two
after the decimal. That is what we care about here. So this anti-log that a calculator gives
needs two significant figures overall. I need to round this to 1.8 times ten to the minus eleventh.
Look at our third and fourth examples: These are both something .32 The digits before the decimal point
are vastly different. But, in the anti-log, you are seeing a calculator gives the same
coefficients, but drastically different exponents. So you can see that the digits before the decimal
point are controlling the exponents. Digits after the decimal point are controlling the coefficient.
But these both have two significant figures after the decimal. Yes, this has three significant figures
overall. Yes, this has five overall. We are looking at how many are after the decimal point. So these
numbers both have two significant figures overall. So the first one, we need to round this to 4.8
times 10 to the minus fifth and the second one, we need to round to 4.8 times 10 to the minus 109.
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