Video Transcript
Use a calculator to find the value
of 𝑥 for which two to the power of two minus five 𝑥 divided by five to the power
of 𝑥 plus three is equal to two. Give your answer correct to two
decimal places.
In this question, we’re given an
equation involving exponents, and we need to solve this equation for the value of
𝑥. We’re told we’re allowed to use a
calculator for this question. We need to give our answer correct
to two decimal places.
Since this is an equation which
involves exponential functions, this should remind us of using logarithms because,
remember, logarithmic functions are the inverse functions of exponential
functions. There’re several different ways of
doing this; we’ll only go through one of these. First, in our equation, notice two
of our terms are two raised to some power. In particular, two is equal to two
to the first power. So, we’ll solve this equation by
taking the log base two of both sides. In fact, we could use any logarithm
base; however, using base two will make this equation the simplest.
So we start by taking the log base
two of both sides of the equation. This gives us the log base two of
two to the power of two minus five 𝑥 divided by five to the power of 𝑥 plus three
is equal to the log base two of two. And although we could evaluate the
log base two of two by using a calculator, this is not necessary. Remember, we define logarithmic
functions to be the inverse of their corresponding exponential functions. In other words, because 𝑎 to the
power of one is equal to 𝑎. We must have the log base 𝑎 of 𝑎
is equal to one, where 𝑎 is a positive number not equal to one.
So, we can simplify the right-hand
side of this equation. The log base two of two is just
equal to one. We now want to simplify the
left-hand side of this equation. And to do this, we need to notice
we’re taking the logarithm of a quotient. This means we can simplify this by
using the following rule, which is often called the quotient rule for
logarithms. The logarithm of a quotient is
equal to the difference of the logarithms. In other words, the log of 𝑎 over
𝑏 is equal to the log of 𝑎 minus the log of 𝑏.
And it’s worth pointing out here,
although we’ve written this as the log base 10, this is actually true for any
logarithm base. So, we can apply this to the
left-hand side of our equation. Instead of taking the logarithm of
this quotient, we’ll take the difference of the logarithms. Doing this, we get the log base two
of two to the power of two minus five 𝑥 minus the log base two of five to the power
of 𝑥 plus three is equal to one. But we still can’t solve this for
our value of 𝑥. Instead, we now need to notice
we’re now taking logarithms of power functions. And we know something which will
help us simplify the logarithm of a power function.
We know the log base 𝑎 of 𝑥 to
the power of 𝑛 is equal to 𝑛 times the log base 𝑎 of 𝑥. This is often called the power rule
for logarithms. And all this says is whenever we’re
taking the logarithm of a power function, we can instead multiply it by our
exponent. Let’s now apply this to both of our
terms.
In our first term, we’re now
instead going to multiply it by our exponent of two. Doing this, we get two minus five
𝑥 multiplied by the log base two of two. We’ll now do the same for our
second term; we’re instead going to multiply it by our exponent of five. This means we’re now subtracting 𝑥
plus three multiplied by the log base two of five. And remember, this is all equal to
one. And now, this is in a form which we
can start to solve. However, there is one thing worth
noticing: the log base two of two is equal to one. So, in fact, this term just
simplified to give us two minus five 𝑥. And there’s a reason for this.
Remember, logarithmic functions and
exponential functions are inverses. So whenever we’re taking the log
base two of two raised to some power, we’re just going to end up with the power. So in fact, we could have skipped
using the power rule for this term altogether. So, we’ll just write our first term
as two minus five 𝑥.
Next, remember, we’re solving for
𝑥. Let’s distribute the log base two
of five over our parentheses. Distributing the log base two of
five and the negative over our parentheses, we get negative 𝑥 times the log base
two of five minus three times the log base two of five. And now this is just an equation we
need to solve for 𝑥. To do this, we need to collect like
terms on both sides of our equation. We’ll do this by adding three times
the log base two of five to both sides and subtracting two. This gives us negative five 𝑥
minus 𝑥 times the log base two of five is equal to one plus three times the log
base two of five minus two.
And on the right-hand side, we can
simplify one minus two is equal to negative one. We want to solve this equation for
𝑥. So on the left-hand side of our
equation, we’ll take out the shared factor of 𝑥. Doing this, we get 𝑥 multiplied by
negative five minus the log base two of five. Finally, all we need to do is
divide both sides of this equation through by negative five minus the log base two
of five. Doing this, we get 𝑥 is equal to
three times the log base two of five minus one all divided by negative five minus
the log base two of five. And if we use our calculator to
evaluate this expression and write our answer to two decimal places, we get negative
0.81, which is our final answer.
Therefore, we were able to find the
value for 𝑥, which solves the equation two to the power of two minus five 𝑥
divided by five to the power of 𝑥 plus three is equal to two. To two decimal places, this was
negative 0.81.
