We want to find the value of each point
on the log scale.
Notice how this looks like a regular number line,
but the units are not ones.
The units are common log X
which would be log base 10 of X,
which is why this is called a log scale.
Sometimes we also see log scales written in exponential form
as we see in this example
which we'll take a look at in another video.
Before we find the value of these four points though,
for a quick review a logarithmic scale is a nonlinear scale
used when there is a large difference or large range
between quantities.
Common logarithmic scales are used to measure the following.
The strength of an earthquake using
the Richter magnitude scale,
the loudness of sound using the decibel scale,
as well as how basic or acidic a solution is
using the pH scale.
All of these scales are logarithmic scales.
Going back to our example, let's start with point A.
Notice how point A is plotted on negative three,
and because the scale is common log X,
this means the common log of X must equal negative three.
X should be the value of point A.
To find the value of point A
we'll write this log equation as an exponential equation.
For a quick review,
if we have log base B of C equals A,
B is the base,
A is the exponent,
and C is the number.
Again because we have common log X
we know this is log base 10 of X.
In exponential form we'd have the base 10
raised to the power of negative three
must equal X or X equals 10 to the negative three.
Another way to form the exponential equation
from the log equation
is to start with the base and work around the equal sign.
Notice how if we start with the base of 10,
we can form the exponential equation by saying 10
raised to the power of negative three
must equal X.
Now to find the value of point A which equals X,
let's first write 10 to the negative three
as one over 10 to the positive three,
which is equal to one one-thousandth
which means point A has a value of one one-thousandth.
Of course we can check this on the calculator.
Notice how 10 raised to the power of negative three
as a decimal is .001.
To convert to a fraction we can press math,
enter, enter.
Now let's take a look at point B.
Notice point B is plotted at negative 1.5
which means common log of X
equals negative 1.5
and X would be the value of point B.
Now we'll write the exponential equation.
Again we have common log which is log base 10,
so the exponential equation would be 10
raised to the power of negative 1.5
must equal X
or X equals 10 raised to the power of negative 1.5.
We can also form this equation by starting with the base
and working around the equal sign.
10 to the power of negative 1.5 must equal X.
Now here we could write 10 to the power of negative 1.5
as one over 10 to the power of 1.5,
but either way here we'll have to get
a decimal approximation.
Going to the calculator we'll just enter 10
raised to the power of negative 1.5.
10 raised to the power of negative 1.5
and enter.
If we round to four decimal places
this would be approximately 0.0316.
Now point C.
Notice point C is plotted at 2.5,
which means common log of X
equals 2.5 where X would be the value of point C.
Again common log is log base 10,
so the exponential equation would be 10
raised to the power of 2.5 equals X,
or if we want X equals 10 raised to the power of 2.5.
Working around the equal sign we'd have 10
raised to the power of 2.5 must equal X.
Here because the exponent is 2.5,
we will have to get a decimal approximation.
Back to the calculator.
10 raised to the power of 2.5
to four decimal places would be approximately
316.2278.
Finally for point D,
notice point D is plotted on four
which means common log X equals four,
where X would be the value of point D.
Common log again is log base 10,
so the exponential equation would be 10
raised to the power of four must equal X,
or X equals 10 to the fourth.
Again working around the equal sign
we have 10 to the fourth must equal X.
10 to the fourth is equal to 10,000
and therefore point D has a value of 10,000.
Of course we can check this.
10 to the fourth equals 10,000.
I hope you found this helpful.