Video Transcript
Find the value of 𝑥 for which
three divided by six to the power of three 𝑥 is equal to two. Give your answer to the nearest
hundredth.
We’re given an equation in terms of
𝑥. And we need to determine the value
of 𝑥 which makes this equation true. And we need to give this value of
𝑥 to the nearest hundredth.
To do this, let’s start by looking
at the equation given to us. We see it’s not easy to rearrange
this equation for 𝑥 directly. So, to help us answer this
question, we need to notice something interesting about the equation given. In the denominator of the fraction
on the left-hand side, we can see we have six to the power of three 𝑥. This is an exponential
function. So we can try and solve this
equation by using logarithms. And there’s several different ways
we could do this. For example, we could just directly
take logarithms of both sides of this equation. We would then need to use the
quotient rule for logarithms. However, it’s always worth trying
to simplify our equation before we take logs of both sides.
So instead we’re going to try and
simplify this equation. We’ll start by multiplying both
sides of our equation through by six to the power of three 𝑥. And by doing this, we get that
three is equal to two multiplied by six to the power of three 𝑥. Now, once again, we could take the
logs of both sides of this equation, and then we would need to use the product rule
for logarithms. However, we can simplify this even
further. We can divide both sides of our
equation through by two. This then gives us that three over
two is equal to six to the power of three 𝑥.
Now, we’re going to need to take
logarithms of both sides of this equation. And there’s a lot of different
options for which logarithm base we could choose. For example, we could use logarithm
base 10. However, on the right-hand side of
our equation, we see we’re raising six to the power of three 𝑥. And because of this, we’re going to
take log base six of both sides of the equation, because remember logarithmic
functions are the inverse functions of exponential functions. So, by choosing log base six, we’re
going to make our equation the most simple. So, by taking the log base six of
both sides of the equation, we have the log base six of three over two is equal to
the log base six of six to the power of three 𝑥.
And there’s a few different ways of
simplifying the expression on the right-hand side of this equation. One way of doing this is noticing
we’re taking the logarithm of an exponential function. And we recall we can simplify this
by using the following rule. The log base 𝑎 of 𝑥 to the power
of 𝑛 is equal to 𝑛 multiplied by the log base 𝑎 of 𝑥. This is sometimes called the power
rule for logarithms. And all this tells us is whenever
we’re taking the logarithm of an exponential function, we can instead just multiply
it by our power. So, by doing this, we get the log
base six of three over two is equal to three 𝑥 multiplied by the log base six of
six.
Now we could just rearrange this
equation for 𝑥 and then solve this by using our calculator. However, there is something worth
pointing out about this. We have the log base six of
six. And if we were to try and evaluate
this by using our calculator, we would see that it’s equal to one. And this is because six to the
power of one is just equal to six. Then, all we need to remember is
logarithmic functions are the inverse of exponential functions. So, when we say the log base six of
six, we want to know which power do we need to raise six to to get our value of
six. And this is of course just equal to
one. And of course this is true for a
lot of different values. The log base 𝑎 of 𝑎 is equal to
one for any positive value of 𝑎 not equal to one. So we’ll use this to simplify our
equation. This just simplifies to give us the
log base six of three over two is equal to three 𝑥.
Finally, we can just rearrange this
equation for 𝑥. We’ll divide both sides of our
equation through by three. So, by doing this and switching the
left- and right-hand side of our equation, we get 𝑥 is equal to one-third times the
log base six of three over two. And if we were to write this into
our calculator, we would get that 𝑥 is equal to 0.07543, and this decimal expansion
continues.
But remember the question wants us
to give our answer to the nearest hundredth. The nearest hundredth is same to
the nearest two decimal places. So, to round this to the nearest
two decimal places, we need to check the third decimal place to see if we round up
or round down. And the digit in the third decimal
place is equal to five. Therefore, we need to round this
value up. So we round this value up. And we get to the nearest hundredth
𝑥 is equal to 0.08, which is our final answer.
Therefore, we were able to find the
value of 𝑥 which solves the equation three divided by six to the power of three 𝑥
is equal to two by using logarithms. To the nearest hundredth, 𝑥 is
equal to 0.08.
