We want to plot the given values
on the log scale.
Looking at the number line, notice how at each tick mark
we have a power of 10.
And as we move from the left to the right at each tick mark,
we increase by a factor of 10 and the exponent
increases by one.
And this is why this is a log scale,
or more specifically a common log scale.
So to determine where to plot point A,
which has a value of 4,000,
we need to solve the equation 10
raised to the power of x equals 4,000.
Notice how if we can determine the value of x,
we can determine where to plot point A on the log scale.
So to solve this equation for x,
we'll write the equivalent log equation.
So for a quick review, if we have the exponential equation B
raised to the power of A equals C,
the equivalent log equation would be 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 10 raised
to the power of x equals 4,000.
Well we know we'd have log.
And because the base is 10, we have log base 10.
The exponent is x, so the log is going to equal x.
And because 10 to the x equals 4,000,
the number is 4,000.
So we have log base 10 of 4,000.
And here because we have log base 10,
this is common log, which means we can leave off the base.
So we have the exponent x is equal to common log 4,000.
And we can approximate common log 4,000 on the calculator.
So enter common log 4,000.
Closed parenthesis and Enter.
And that's rounded four decimal places.
Notice how x or common log 4,000 is going to be
approximately 3.6021.
So now we know that 3.6021 is
approximately equal to common log 4,000.
Which means x is approximately 3.6021.
And therefore learning the equivalent exponential
equation, now that we know x, we can say that 10
raised to the power of 3.6021
is approximately 4,000.
Let's go ahead and check this on the calculator.
If we enter 10 raised to the power of 3.6021
and then Enter, notice how we get approximately 4,000.
It's a little bit more than 4,000 because we did round up
to get our decimal approximation.
But now we can determine where to plot point A.
10 raised to the power of 3.6021 would be
between 10 to the third and 10 to the fourth.
A little bit closer to 10 to the fourth.
Let's go ahead and plot point A here.
Which again has a value of 4,000.
And now to determine where to plot point B,
which has a value of 0.002.
So to determine where to plot point B,
we want to solve the exponential equation 10 raised
to the power of x equals 0.002.
So to solve for x, write the equivalent log equation.
So because we have base 10, we know we're going to have
log base 10, because the exponent is x,
the logarithm is going to equal x.
And because 10 to the x equals this decimal,
we know we'll have log base 10 of 0.002.
And now with a calculator to approximate this common
logarithm, which would be the value of x.
And again, because we have log base 10,
we can leave off the base because this is common log.
So we'll enter common log .002 closed parenthesis, Enter.
And if we round to four decimal places, notice how we have
a seven in the fifth decimal place,
which indicates you round up.
And because we have a nine here, this is going to round to
approximately negative 2.6990.
So we know common log of 0.002 is approximately
negative 2.6990.
Which also tells us that 10 raised to the power of
negative 2.6990 is approximately 0.002.
And again, let's go ahead and verify this on the calculator.
So 10 raised to the power of negative 2.6990
should give us approximately 0.002.
And notice how it does.
So to plot point B, we need to determine where 10
raised to the power of negative 2.6990 would be
on the log scale.
Well this would be between 10 to the negative two
and 10 to the negative three, but closer to
10 to the negative three.
So let's plot point B here.
Which again has a value of 0.002.
Okay I hope you found this helpful.