We want to find the value
of each point on the logarithmic scale.
Looking at the scale, the number line,
notice how each tick mark represents
a power of 10, which is
why this is a log scale,
or more specifically, a common log scale.
So here, this scale is
written in exponential form,
but as we saw in the previous example,
the scale can also be in log form
as we see here the scale is common log x.
So, both of these number
lines are log scales,
but for this example, we're
going to be considering
the number line where the
scale is exponential form.
So, let's first consider
the value of point A.
If we let x be equal to
the value of point A,
notice how x would be equal to 10
raised to the power of negative three.
To simplify this by
hand, we can write this
as a fraction with the denominator of one.
We move this down to the denominator,
it would change the sign of the exponent,
so we'd have x equals
one over 10 to the third,
which is equal to one one thousandth.
Of course, we can check this
on the calculator if we wish.
We enter 10 raised to the
power of negative three, enter,
which gives us the decimal value
of point zero zero one, or one thousandth.
To convert to a fraction,
we press math, enter, enter.
So, the value of point
A is one one thousandth,
but before we go on to point B,
let's go back to the
first exponential equation
x equals 10 raised to the
power of negative three
and write this as an
equivalent log equation
to emphasis why this is a log scale.
So, for a quick review, if we
have the exponential equation
B raised to the power of A equals C,
the equivalent log equation
is log base B of C equals A,
where B is the base,
A is the exponent,
and C is the number.
So, the equivalent log
equation to x equals 10
raised to the power of
negative three would be...
Well first, the base is 10, so
we know we'd have log base 10
because the exponent is negative three,
we know the log is going
to equal negative three.
And because 10 to the
negative three equals x,
we have log base 10 of
x equals negative three.
But because we have log
base 10, this is common log,
and we can leave off the base 10.
So, notice how if we
take a look at point A,
the exponent on 10 is negative three,
which is what common log x is equal to.
So, looking at the number line,
notice how as you move to
the right, each tick mark
would pick up a factor of 10,
and therefore the exponent,
or the common log of x, increases by one,
which, again, is why this
is called a log scale.
Now let's consider point B.
If we let x be equal to
the value of point B,
notice how point B is between
10 to the negative one
and 10 to the zero, and therefore,
x would be equal to 10
raised to the power of negative one half
or negative zero point
five, which we could write
as one over 10 raised to the power
of positive zero point five,
but let's also get our
decimal approximation.
And we'll go ahead and just enter 10
raised to the power of
negative point five, enter,
and this will not convert
to a nice fraction,
so I'll round to four decimal places.
This will be approximately
zero point three one six two.
So, this is the value of point B.
Let's also write this exponential equation
as a log equation.
We'd have log base 10, or common log, of x
equals negative zero point five.
So going back to the
number line, notice how
it's not shown on the
scale, but this would be 10
raised to the power of
negative zero point five,
and notice how common log x is equal
to negative zero point five,
which is the exponent here at point B.
And now we'll consider point C.
If we let x be equal to
the value of point C,
notice how we'd have x
equal 10 to the second,
which is equal to 100.
And here, the equivalent log equation
would be the common log of x
equals the exponent of two.
And finally for point D,
if we let x be equal to
the value of point D,
notice how point D is
between 10 to the third
and 10 to the fourth, halfway
between three and four
would be three point five, and therefore,
the value of D would be x equals 10
to the power of three point five,
and here we'll have to get
our decimal approximation,
so we'll go back to the calculator.
We would have 10 raised to the power
of three point five, enter.
To four decimal places we'd have
approximately 3,162.2777,
which is the approximate value of point D.
And finally, one more time let's write
the equivalent log equation.
We'd have common log of x equals
the exponent of three point five,
which again, if we labelled point D
it would be 10 raised to the
power of three point five.
So, once again, notice
how the common log of x
equals three point five,
which is the exponent
at point D.
Okay, I hope you found this helpful.
