Video Transcript
If 𝑎 is in the open interval from
one to 1,000, what interval contains the logarithm of 𝑎?
In this question, we’re told that
the constant value of 𝑎 lies in the open interval from one to 1,000. And we need to use this to
determine in which interval would be the logarithm of 𝑎. To answer this question, we’ll
start by recalling when the base of a logarithm is not told to us, this means its
base is 10, so this means the logarithm base 10 of 𝑎. Next, since we’re told that 𝑎 lies
in the open interval from one to 1,000, we recall this means that 𝑎 is bigger than
one and less than 1,000. Therefore, we need to determine
information about the log base 10 of 𝑎 given that 𝑎 is between one and 1,000. We’ll do this by using a graph.
We’ll sketch the curve 𝑦 is equal
to the log base 10 of 𝑥. And we recall since the base 10 is
greater than one, this will be an increasing logarithmic curve. There will be an 𝑥-intercept at
one and the 𝑦-axis will act as a vertical asymptote. This gives us the following
sketch. We need to determine the possible
values of the log base 10 of 𝑎 given that 𝑎 is between one and 1,000. And to do this, we recall in the
graph of a function the 𝑥-coordinates of the points on the curve tell us the input
values and the 𝑦-coordinates tell us the corresponding outputs of the function.
For example, we can see that the
point with coordinates one, zero lies on our curve. This tells us that the log base 10
of one is equal to zero. Another way of thinking about this
is the 𝑦-coordinate of the point with 𝑥-coordinate one is zero. Since our value of 𝑎 must be less
than 1,000, let’s see what happens when we input 1,000 into the function. To do this, we recall the following
fact about logarithms. For any positive real number 𝑏 not
equal to one, the log base 𝑏 of 𝑏 to the power of 𝑛 is just equal to 𝑛. This is because the logarithm base
𝑏 function is the inverse of the exponential function 𝑏 to the power of 𝑛. We can use this to evaluate the log
base 10 of 1,000, since 1,000 is 10 cubed.
We get the log base 10 of 10 cubed
is the exponent value of three. We can then add this onto our
diagram. It’s the 𝑦-coordinate of the point
on our curve of 𝑥-coordinate 1,000. Now, we can notice something
interesting in our diagram. If our input values of 𝑥 lie
between one and 1,000, then the output values of 𝑦 will lie between zero and
three. Therefore, we’ve shown that zero is
less than the log base 10 of 𝑎 is less than three, where we use strict inequalities
because we’re not allowed 𝑎 is equal to one and we’re not allowed 𝑎 is equal to
1,000.
And the question wants us to write
this in interval notation. And we can write this by saying the
log base 𝑎 must be in the open interval from zero to three, which is our final
answer. Therefore, we were able to show if
𝑎 is in the open interval from one to 1,000, then the log base 10 of 𝑎 is in the
open interval from zero to three.
